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Core 3 - HackMD
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<div id="doc" class="markdown-body container-fluid"><h1 id="core-3"><a class="anchor hidden-xs" href="#core-3" title="core-3"><span class="octicon octicon-link"></span></a>Core 3</h1><h2 id="functions"><a class="anchor hidden-xs" href="#functions" title="functions"><span class="octicon octicon-link"></span></a>Functions</h2><p>A function is a one to one mapping</p><h3 id="domain-and-range-of-a-function"><a class="anchor hidden-xs" href="#domain-and-range-of-a-function" title="domain-and-range-of-a-function"><span class="octicon octicon-link"></span></a>Domain and range of a function</h3><p>The domain of a function is the set of valid input values</p><p>A function is defined by both a rule and a domain</p><h3 id="inverse-of-a-function"><a class="anchor hidden-xs" href="#inverse-of-a-function" title="inverse-of-a-function"><span class="octicon octicon-link"></span></a>Inverse of a function</h3><p>An inverse function is such that <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-1-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><msup><mi>f</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>f</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>1</mn></mrow></msup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1" style="width: 11.017em; display: inline-block;"><span style="display: inline-block; position: relative; width: 9.019em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1009.02em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2"><span class="mi" id="MathJax-Span-3" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="msubsup" id="MathJax-Span-4"><span style="display: inline-block; position: relative; width: 1.437em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-5" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.515em;"><span class="texatom" id="MathJax-Span-6"><span class="mrow" id="MathJax-Span-7"><span class="mo" id="MathJax-Span-8" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-9" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-10" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-11" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-12" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-13" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="msubsup" id="MathJax-Span-14" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.437em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-15" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.515em;"><span class="texatom" id="MathJax-Span-16"><span class="mrow" id="MathJax-Span-17"><span class="mo" id="MathJax-Span-18" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-19" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mi" id="MathJax-Span-20" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-21" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-22" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-23" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-24" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-25" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.378em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><msup><mi>f</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>f</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn></mrow></msup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi></math></span></span><script type="math/tex" id="MathJax-Element-1">ff^{-1} (x) = f^{-1} f(x) = x</script></span></p><p>For a function to have a valid inverse function, it must be a one to one mapping</p><h2 id="transformations-and-the-modulus-function"><a class="anchor hidden-xs" href="#transformations-and-the-modulus-function" title="transformations-and-the-modulus-function"><span class="octicon octicon-link"></span></a>Transformations and the modulus function</h2><h3 id="translations"><a class="anchor hidden-xs" href="#translations" title="translations"><span class="octicon octicon-link"></span></a>Translations</h3><p>Translating the function <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-45-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1326" style="width: 3.998em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.281em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1003.23em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1327"><span class="mi" id="MathJax-Span-1328" style="font-family: STIXGeneral; font-style: italic;">y</span><span class="mo" id="MathJax-Span-1329" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-1330" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-1331" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1332" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1333" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-45">y = f(x)</script></span> by the vector <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-46-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable><mo>]</mo></mrow></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1334" style="width: 2.257em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.847em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(2.513em 1001.64em 5.279em -999.997em); top: -4.147em; left: 0em;"><span class="mrow" id="MathJax-Span-1335"><span class="mrow" id="MathJax-Span-1336"><span class="mo" id="MathJax-Span-1337" style="vertical-align: -0.561em;"><span style="font-family: STIXSizeThreeSym;">[</span></span><span class="mtable" id="MathJax-Span-1338" style="padding-right: 0.156em; padding-left: 0.156em;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px;"><span style="position: absolute; clip: rect(2.718em 1000.46em 4.921em -999.997em); top: -3.993em; left: 0em;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mtd" id="MathJax-Span-1339"><span class="mrow" id="MathJax-Span-1340"><span class="mi" id="MathJax-Span-1341" style="font-family: STIXGeneral; font-style: italic;">a</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.46em 4.152em -999.997em); top: -3.225em; left: 50%; margin-left: -0.254em;"><span class="mtd" id="MathJax-Span-1342"><span class="mrow" id="MathJax-Span-1343"><span class="mi" id="MathJax-Span-1344" style="font-family: STIXGeneral; font-style: italic;">b</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1345" style="vertical-align: -0.561em;"><span style="font-family: STIXSizeThreeSym;">]</span></span></span></span><span style="display: inline-block; width: 0px; height: 4.152em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.247em; border-left: 0px solid; width: 0px; height: 3.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable><mo>]</mo></mrow></math></span></span><script type="math/tex" id="MathJax-Element-46">\begin{bmatrix} a \\ b \end{bmatrix}</script></span> gives <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-47-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>&#x2212;</mo><mi>a</mi><mo stretchy="false">)</mo><mo>+</mo><mi>b</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1346" style="width: 8.148em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.662em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1006.61em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1347"><span class="mi" id="MathJax-Span-1348" style="font-family: STIXGeneral; font-style: italic;">y</span><span class="mo" id="MathJax-Span-1349" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-1350" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-1351" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1352" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1353" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-1354" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">a</span><span class="mo" id="MathJax-Span-1355" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1356" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-1357" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">b</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>a</mi><mo stretchy="false">)</mo><mo>+</mo><mi>b</mi></math></span></span><script type="math/tex" id="MathJax-Element-47">y = f(x-a) +b</script></span></p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-4-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>k</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-50" style="width: 4.562em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.742em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1003.69em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-51"><span class="mi" id="MathJax-Span-52" style="font-family: STIXGeneral; font-style: italic;">y</span><span class="mo" id="MathJax-Span-53" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-54" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-55" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-56" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-57" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-58" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>k</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-4">y = kf(x)</script></span> is a stretch of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-5-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-59" style="width: 3.998em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.281em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1003.23em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-60"><span class="mi" id="MathJax-Span-61" style="font-family: STIXGeneral; font-style: italic;">y</span><span class="mo" id="MathJax-Span-62" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-63" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-64" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-65" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-66" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-5">y = f(x)</script></span> by a factor of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-6-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-67" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-68"><span class="mi" id="MathJax-Span-69" style="font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-6">k</script></span> in the <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-7-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-70" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-71"><span class="mi" id="MathJax-Span-72" style="font-family: STIXGeneral; font-style: italic;">y</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math></span></span><script type="math/tex" id="MathJax-Element-7">y</script></span> direction</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-8-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mfrac><mi>x</mi><mi>k</mi></mfrac><mo>)</mo></mrow></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-73" style="width: 4.869em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.998em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1003.85em 3.537em -999.997em); top: -2.969em; left: 0em;"><span class="mrow" id="MathJax-Span-74"><span class="mi" id="MathJax-Span-75" style="font-family: STIXGeneral; font-style: italic;">y</span><span class="mo" id="MathJax-Span-76" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-77" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mrow" id="MathJax-Span-78" style="padding-left: 0.207em;"><span class="mo" id="MathJax-Span-79" style="vertical-align: -0.202em;"><span style="font-family: STIXSizeOneSym;">(</span></span><span class="mfrac" id="MathJax-Span-80"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.537em 1000.31em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.151em;"><span class="mi" id="MathJax-Span-81" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.31em 4.152em -999.997em); top: -3.584em; left: 50%; margin-left: -0.151em;"><span class="mi" id="MathJax-Span-82" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.46em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.464em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-83" style="vertical-align: -0.202em;"><span style="font-family: STIXSizeOneSym;">)</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.974em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.559em; border-left: 0px solid; width: 0px; height: 1.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mfrac><mi>x</mi><mi>k</mi></mfrac><mo>)</mo></mrow></math></span></span><script type="math/tex" id="MathJax-Element-8">y = f\left(\frac x k \right)</script></span> is a stretch y a factor of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-9-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-84" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-85"><span class="mi" id="MathJax-Span-86" style="font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-9">k</script></span> in the <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-10-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-87" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-88"><span class="mi" id="MathJax-Span-89" style="font-family: STIXGeneral; font-style: italic;">y</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math></span></span><script type="math/tex" id="MathJax-Element-10">y</script></span> direction</p><p>Reflecting <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-188-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-7183" style="width: 3.998em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.281em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1003.23em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-7184"><span class="mi" id="MathJax-Span-7185" style="font-family: STIXGeneral; font-style: italic;">y</span><span class="mo" id="MathJax-Span-7186" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-7187" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-7188" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-7189" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-7190" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-188">y=f(x)</script></span> in the y axis gives <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-189-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mo>&#x2212;</mo><mi>x</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-7191" style="width: 4.869em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.998em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1003.95em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-7192"><span class="mi" id="MathJax-Span-7193" style="font-family: STIXGeneral; font-style: italic;">y</span><span class="mo" id="MathJax-Span-7194" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-7195" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-7196" style="font-family: STIXGeneral;">(</span><span class="mo" id="MathJax-Span-7197" style="font-family: STIXGeneral;">−</span><span class="mi" id="MathJax-Span-7198" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-7199" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-189">y=f(-x)</script></span></p><p>The modulus of a function is given by:<br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-211-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>=</mo><mrow><mo>{</mo><mtable columnalign="left left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd><mtd><mspace width="1em" /><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>&#x2265;</mo><mn>0</mn></mtd></mtr><mtr><mtd><mo>&#x2212;</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd><mtd><mspace width="1em" /><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>&lt;</mo><mn>0</mn></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-8064" style="width: 15.32em; display: inline-block;"><span style="display: inline-block; position: relative; width: 12.554em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(2.513em 1012.55em 5.279em -999.997em); top: -4.147em; left: 0em;"><span class="mrow" id="MathJax-Span-8065"><span class="texatom" id="MathJax-Span-8066"><span class="mrow" id="MathJax-Span-8067"><span class="mo" id="MathJax-Span-8068" style="font-family: STIXVariants;">|</span></span></span><span class="mi" id="MathJax-Span-8069" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-8070" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-8071" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-8072" style="font-family: STIXGeneral;">)</span><span class="texatom" id="MathJax-Span-8073"><span class="mrow" id="MathJax-Span-8074"><span class="mo" id="MathJax-Span-8075" style="font-family: STIXVariants;">|</span></span></span><span class="mo" id="MathJax-Span-8076" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mrow" id="MathJax-Span-8077" style="padding-left: 0.31em;"><span class="mo" id="MathJax-Span-8078" style="vertical-align: -0.561em;"><span style="font-family: STIXSizeThreeSym;">{</span></span><span class="mtable" id="MathJax-Span-8079" style="padding-right: 0.156em; padding-left: 0.156em;"><span style="display: inline-block; position: relative; width: 7.841em; height: 0px;"><span style="position: absolute; clip: rect(2.462em 1002.15em 5.126em -999.997em); top: -3.993em; left: 0em;"><span style="display: inline-block; position: relative; width: 2.205em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1001.49em 4.357em -999.997em); top: -4.659em; left: 0em;"><span class="mtd" id="MathJax-Span-8080"><span class="mrow" id="MathJax-Span-8081"><span class="mi" id="MathJax-Span-8082" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-8083" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-8084" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-8085" style="font-family: STIXGeneral;">)</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1002.15em 4.357em -999.997em); top: -3.225em; left: 0em;"><span class="mtd" id="MathJax-Span-8095"><span class="mrow" id="MathJax-Span-8096"><span class="mo" id="MathJax-Span-8097" style="font-family: STIXGeneral;">−</span><span class="mi" id="MathJax-Span-8098" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-8099" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-8100" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-8101" style="font-family: STIXGeneral;">)</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.462em 1004.46em 5.126em -999.997em); top: -3.993em; left: 3.332em;"><span style="display: inline-block; position: relative; width: 4.46em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1004.46em 4.357em -999.997em); top: -4.659em; left: 0em;"><span class="mtd" id="MathJax-Span-8086"><span class="mrow" id="MathJax-Span-8087"><span class="mspace" id="MathJax-Span-8088" style="height: 0em; vertical-align: 0em; width: 1.13em; display: inline-block; overflow: hidden;"></span><span class="mi" id="MathJax-Span-8089" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-8090" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-8091" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-8092" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-8093" style="font-family: STIXGeneral; padding-left: 0.31em;">≥</span><span class="mn" id="MathJax-Span-8094" style="font-family: STIXGeneral; padding-left: 0.31em;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1004.46em 4.357em -999.997em); top: -3.225em; left: 0em;"><span class="mtd" id="MathJax-Span-8102"><span class="mrow" id="MathJax-Span-8103"><span class="mspace" id="MathJax-Span-8104" style="height: 0em; vertical-align: 0em; width: 1.13em; display: inline-block; overflow: hidden;"></span><span class="mi" id="MathJax-Span-8105" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-8106" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-8107" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-8108" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-8109" style="font-family: STIXGeneral; padding-left: 0.31em;"><</span><span class="mn" id="MathJax-Span-8110" style="font-family: STIXGeneral; padding-left: 0.31em;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-8111"></span></span></span><span style="display: inline-block; width: 0px; height: 4.152em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.247em; border-left: 0px solid; width: 0px; height: 3.128em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>=</mo><mrow><mo>{</mo><mtable columnalign="left left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd><mtd><mspace width="1em"></mspace><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≥</mo><mn>0</mn></mtd></mtr><mtr><mtd><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd><mtd><mspace width="1em"></mspace><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo><</mo><mn>0</mn></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-211">|f(x)| = \left\{
\begin{array}{ll}
f(x) & \quad f(x) \geq 0 \\
-f(x) & \quad f(x) < 0
\end{array}
\right.
</script></span></p><h2 id="inverse-trigonometric-functions"><a class="anchor hidden-xs" href="#inverse-trigonometric-functions" title="inverse-trigonometric-functions"><span class="octicon octicon-link"></span></a>Inverse trigonometric functions</h2><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-146-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>asin</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-6813" style="width: 3.332em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.718em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.67em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-6814"><span class="mi" id="MathJax-Span-6815" style="font-family: STIXGeneral;">asin</span><span class="mo" id="MathJax-Span-6816"></span><span class="mo" id="MathJax-Span-6817" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-6818" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-6819" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>asin</mi><mo></mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-146">\operatorname{asin}(x)</script></span> has domain <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-147-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x2212;</mo><mn>1</mn><mo>&#x2264;</mo><mi>x</mi><mo>&#x2264;</mo><mn>1</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-6820" style="width: 5.791em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.716em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1004.61em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-6821"><span class="mo" id="MathJax-Span-6822" style="font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-6823" style="font-family: STIXGeneral;">1</span><span class="mo" id="MathJax-Span-6824" style="font-family: STIXGeneral; padding-left: 0.31em;">≤</span><span class="mi" id="MathJax-Span-6825" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-6826" style="font-family: STIXGeneral; padding-left: 0.31em;">≤</span><span class="mn" id="MathJax-Span-6827" style="font-family: STIXGeneral; padding-left: 0.31em;">1</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mn>1</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-147">-1 \leq x \leq 1</script></span> and range of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-148-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x2212;</mo><mfrac><mi>&#x03C0;</mi><mn>2</mn></mfrac><mo>&#x2264;</mo><mi>asin</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>&#x2264;</mo><mfrac><mi>&#x03C0;</mi><mn>2</mn></mfrac></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-6828" style="width: 9.07em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.431em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.693em 1007.43em 3.076em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-6829"><span class="mo" id="MathJax-Span-6830" style="font-family: STIXGeneral;">−</span><span class="mfrac" id="MathJax-Span-6831"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.202em;"><span class="mi" id="MathJax-Span-6832" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">π<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.635em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-6833" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.51em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.515em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-6834" style="font-family: STIXGeneral; padding-left: 0.31em;">≤</span><span class="mi" id="MathJax-Span-6835" style="font-family: STIXGeneral; padding-left: 0.31em;">asin</span><span class="mo" id="MathJax-Span-6836"></span><span class="mo" id="MathJax-Span-6837" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-6838" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-6839" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-6840" style="font-family: STIXGeneral; padding-left: 0.31em;">≤</span><span class="mfrac" id="MathJax-Span-6841" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.202em;"><span class="mi" id="MathJax-Span-6842" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">π<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.635em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-6843" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.51em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.515em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.497em; border-left: 0px solid; width: 0px; height: 1.503em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mfrac><mi>π</mi><mn>2</mn></mfrac><mo>≤</mo><mi>asin</mi><mo></mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≤</mo><mfrac><mi>π</mi><mn>2</mn></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-148">-\frac{\pi}{2} \leq \operatorname{asin}(x) \leq \frac{\pi}{2}</script></span></p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-161-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>acos</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-6951" style="width: 3.537em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.871em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.82em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-6952"><span class="mi" id="MathJax-Span-6953" style="font-family: STIXGeneral;">acos</span><span class="mo" id="MathJax-Span-6954"></span><span class="mo" id="MathJax-Span-6955" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-6956" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-6957" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>acos</mi><mo></mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-161">\operatorname{acos}(x)</script></span> has domain <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-162-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x2212;</mo><mn>1</mn><mo>&#x2264;</mo><mi>x</mi><mo>&#x2264;</mo><mn>1</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-6958" style="width: 5.791em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.716em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1004.61em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-6959"><span class="mo" id="MathJax-Span-6960" style="font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-6961" style="font-family: STIXGeneral;">1</span><span class="mo" id="MathJax-Span-6962" style="font-family: STIXGeneral; padding-left: 0.31em;">≤</span><span class="mi" id="MathJax-Span-6963" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-6964" style="font-family: STIXGeneral; padding-left: 0.31em;">≤</span><span class="mn" id="MathJax-Span-6965" style="font-family: STIXGeneral; padding-left: 0.31em;">1</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mn>1</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-162">-1 \leq x \leq 1</script></span> and range of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-163-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>&#x2264;</mo><mi>acos</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>&#x2264;</mo><mfrac><mi>&#x03C0;</mi><mn>2</mn></mfrac></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-6966" style="width: 8.199em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.714em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.693em 1006.71em 3.076em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-6967"><span class="mn" id="MathJax-Span-6968" style="font-family: STIXGeneral;">0</span><span class="mo" id="MathJax-Span-6969" style="font-family: STIXGeneral; padding-left: 0.31em;">≤</span><span class="mi" id="MathJax-Span-6970" style="font-family: STIXGeneral; padding-left: 0.31em;">acos</span><span class="mo" id="MathJax-Span-6971"></span><span class="mo" id="MathJax-Span-6972" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-6973" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-6974" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-6975" style="font-family: STIXGeneral; padding-left: 0.31em;">≤</span><span class="mfrac" id="MathJax-Span-6976" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.202em;"><span class="mi" id="MathJax-Span-6977" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">π<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.635em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-6978" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.51em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.515em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.497em; border-left: 0px solid; width: 0px; height: 1.503em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>acos</mi><mo></mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≤</mo><mfrac><mi>π</mi><mn>2</mn></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-163">0\leq \operatorname{acos}(x) \leq \frac{\pi}{2}</script></span></p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-170-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>atan</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-7033" style="width: 3.384em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.769em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.72em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-7034"><span class="mi" id="MathJax-Span-7035" style="font-family: STIXGeneral;">atan</span><span class="mo" id="MathJax-Span-7036"></span><span class="mo" id="MathJax-Span-7037" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-7038" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-7039" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>atan</mi><mo></mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-170">\operatorname{atan}(x)</script></span> has domain <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-171-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&#x2208;</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi></mrow></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-7040" style="width: 3.076em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.513em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.46em 2.769em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-7041"><span class="mi" id="MathJax-Span-7042" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-7043" style="font-family: STIXGeneral; padding-left: 0.31em;">∈</span><span class="texatom" id="MathJax-Span-7044" style="padding-left: 0.31em;"><span class="mrow" id="MathJax-Span-7045"><span class="mi" id="MathJax-Span-7046" style="font-family: STIXGeneral;">ℝ</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.122em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-171">x \in \mathbb{R}</script></span> and range of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-172-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x2212;</mo><mfrac><mi>&#x03C0;</mi><mn>2</mn></mfrac><mo>&#x2264;</mo><mi>atan</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>&#x2264;</mo><mfrac><mi>&#x03C0;</mi><mn>2</mn></mfrac></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-7047" style="width: 9.224em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.533em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.693em 1007.53em 3.076em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-7048"><span class="mo" id="MathJax-Span-7049" style="font-family: STIXGeneral;">−</span><span class="mfrac" id="MathJax-Span-7050"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.202em;"><span class="mi" id="MathJax-Span-7051" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">π<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.635em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-7052" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.51em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.515em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-7053" style="font-family: STIXGeneral; padding-left: 0.31em;">≤</span><span class="mi" id="MathJax-Span-7054" style="font-family: STIXGeneral; padding-left: 0.31em;">atan</span><span class="mo" id="MathJax-Span-7055"></span><span class="mo" id="MathJax-Span-7056" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-7057" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-7058" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-7059" style="font-family: STIXGeneral; padding-left: 0.31em;">≤</span><span class="mfrac" id="MathJax-Span-7060" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.202em;"><span class="mi" id="MathJax-Span-7061" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">π<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.635em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-7062" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.51em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.515em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.497em; border-left: 0px solid; width: 0px; height: 1.503em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mfrac><mi>π</mi><mn>2</mn></mfrac><mo>≤</mo><mi>atan</mi><mo></mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≤</mo><mfrac><mi>π</mi><mn>2</mn></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-172">-\frac{\pi}{2} \leq \operatorname{atan}(x) \leq \frac{\pi}{2}</script></span></p><h2 id="trig-identities"><a class="anchor hidden-xs" href="#trig-identities" title="trig-identities"><span class="octicon octicon-link"></span></a>Trig identities</h2><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-110-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd /><mtd><mi>sin</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mi>cos</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>=</mo><mn>1</mn></mtd></mtr><mtr><mtd /><mtd><mn>1</mn><mo>+</mo><mi>cot</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>=</mo><mi>cosec</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mtd></mtr><mtr><mtd /><mtd><mn>1</mn><mo>+</mo><mi>tan</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>=</mo><mi>sec</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mtd></mtr></mtable></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2414" style="width: 12.041em; display: inline-block;"><span style="display: inline-block; position: relative; width: 9.839em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(-0.151em 1009.68em 4.101em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-2415"><span class="mtable" id="MathJax-Span-2416" style="padding-right: 0.156em; padding-left: 0.156em;"><span style="display: inline-block; position: relative; width: 9.531em; height: 0px;"><span style="position: absolute; clip: rect(2.513em 1000em 5.689em -999.997em); top: -3.993em; left: 0em;"><span style="display: inline-block; position: relative; width: 0em; height: 0px;"><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -5.325em; right: 0em;"><span class="mtd" id="MathJax-Span-2417"><span class="mrow" id="MathJax-Span-2418"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -3.891em; right: 0em;"><span class="mtd" id="MathJax-Span-2438"><span class="mrow" id="MathJax-Span-2439"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -2.456em; right: 0em;"><span class="mtd" id="MathJax-Span-2459"><span class="mrow" id="MathJax-Span-2460"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.564em 1009.53em 6.765em -999.997em); top: -4.915em; left: 0em;"><span style="display: inline-block; position: relative; width: 9.531em; height: 0px;"><span style="position: absolute; clip: rect(2.974em 1008.51em 4.306em -999.997em); top: -5.325em; left: 0em;"><span class="mtd" id="MathJax-Span-2419"><span class="mrow" id="MathJax-Span-2420"><span class="mi" id="MathJax-Span-2421" style="font-family: STIXGeneral;">sin</span><span class="mo" id="MathJax-Span-2422"></span><span class="mo" id="MathJax-Span-2423" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2424" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-2425"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.31em 4.306em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-2426" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.361em;"><span class="mn" id="MathJax-Span-2427" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-2428" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-2429" style="font-family: STIXGeneral; padding-left: 0.259em;">cos</span><span class="mo" id="MathJax-Span-2430"></span><span class="mo" id="MathJax-Span-2431" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2432" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-2433"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.31em 4.306em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-2434" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.361em;"><span class="mn" id="MathJax-Span-2435" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-2436" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mn" id="MathJax-Span-2437" style="font-family: STIXGeneral; padding-left: 0.31em;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.974em 1009.53em 4.306em -999.997em); top: -3.891em; left: 0em;"><span class="mtd" id="MathJax-Span-2440"><span class="mrow" id="MathJax-Span-2441"><span class="mn" id="MathJax-Span-2442" style="font-family: STIXGeneral;">1</span><span class="mo" id="MathJax-Span-2443" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-2444" style="font-family: STIXGeneral; padding-left: 0.259em;">cot</span><span class="mo" id="MathJax-Span-2445"></span><span class="mo" id="MathJax-Span-2446" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2447" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-2448"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.31em 4.306em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-2449" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.361em;"><span class="mn" id="MathJax-Span-2450" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-2451" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-2452" style="font-family: STIXGeneral; padding-left: 0.31em;">cosec</span><span class="mo" id="MathJax-Span-2453"></span><span class="mo" id="MathJax-Span-2454" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2455" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-2456"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.31em 4.306em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-2457" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.361em;"><span class="mn" id="MathJax-Span-2458" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.974em 1008.61em 4.306em -999.997em); top: -2.456em; left: 0em;"><span class="mtd" id="MathJax-Span-2461"><span class="mrow" id="MathJax-Span-2462"><span class="mn" id="MathJax-Span-2463" style="font-family: STIXGeneral;">1</span><span class="mo" id="MathJax-Span-2464" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-2465" style="font-family: STIXGeneral; padding-left: 0.259em;">tan</span><span class="mo" id="MathJax-Span-2466"></span><span class="mo" id="MathJax-Span-2467" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2468" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-2469"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.31em 4.306em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-2470" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.361em;"><span class="mn" id="MathJax-Span-2471" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-2472" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-2473" style="font-family: STIXGeneral; padding-left: 0.31em;">sec</span><span class="mo" id="MathJax-Span-2474"></span><span class="mo" id="MathJax-Span-2475" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2476" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-2477"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.31em 4.306em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-2478" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.361em;"><span class="mn" id="MathJax-Span-2479" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.921em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -2.184em; border-left: 0px solid; width: 0px; height: 4.941em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd></mtd><mtd><mi>sin</mi><mo></mo><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mi>cos</mi><mo></mo><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>=</mo><mn>1</mn></mtd></mtr><mtr><mtd></mtd><mtd><mn>1</mn><mo>+</mo><mi>cot</mi><mo></mo><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>=</mo><mi>cosec</mi><mo></mo><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mtd></mtr><mtr><mtd></mtd><mtd><mn>1</mn><mo>+</mo><mi>tan</mi><mo></mo><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>=</mo><mi>sec</mi><mo></mo><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-110">\begin{align}
&\sin(x)^2 + \cos(x)^2 = 1 \\
&1 + \cot(x)^2 = \operatorname{cosec}(x)^2 \\
&1 + \tan(x)^2 = \sec(x)^2\end{align}</script></span></p><h2 id="calculus"><a class="anchor hidden-xs" href="#calculus" title="calculus"><span class="octicon octicon-link"></span></a>Calculus</h2><h3 id="log-rules"><a class="anchor hidden-xs" href="#log-rules" title="log-rules"><span class="octicon octicon-link"></span></a>Log rules</h3><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-121-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd /><mtd><mi>ln</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>+</mo><mi>ln</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ln</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd /><mtd><mi>ln</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>&#x2212;</mo><mi>ln</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ln</mi><mo>&#x2061;</mo><mrow><mo>(</mo><mfrac><mi>a</mi><mi>b</mi></mfrac><mo>)</mo></mrow></mtd></mtr><mtr><mtd /><mtd><mi>n</mi><mi>ln</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ln</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><msup><mi>a</mi><mi>n</mi></msup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd /><mtd><msub><mi>log</mi><mi>a</mi></msub><mo>&#x2061;</mo><mi>b</mi><mo>=</mo><mfrac><mrow><msub><mi>log</mi><mi>c</mi></msub><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi>log</mi><mi>c</mi></msub><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mfrac></mtd></mtr></mtable></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3447" style="width: 11.632em; display: inline-block;"><span style="display: inline-block; position: relative; width: 9.531em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(-1.842em 1009.22em 5.74em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-3448"><span class="mtable" id="MathJax-Span-3449" style="padding-right: 0.156em; padding-left: 0.156em;"><span style="display: inline-block; position: relative; width: 9.224em; height: 0px;"><span style="position: absolute; clip: rect(3.281em 1000em 9.07em -999.997em); top: -6.504em; left: 0em;"><span style="display: inline-block; position: relative; width: 0em; height: 0px;"><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -7.067em; right: 0em;"><span class="mtd" id="MathJax-Span-3450"><span class="mrow" id="MathJax-Span-3451"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -5.377em; right: 0em;"><span class="mtd" id="MathJax-Span-3472"><span class="mrow" id="MathJax-Span-3473"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -3.532em; right: 0em;"><span class="mtd" id="MathJax-Span-3496"><span class="mrow" id="MathJax-Span-3497"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -1.586em; right: 0em;"><span class="mtd" id="MathJax-Span-3514"><span class="mrow" id="MathJax-Span-3515"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 6.509em;"></span></span><span style="position: absolute; clip: rect(4.255em 1009.12em 11.683em -999.997em); top: -8.143em; left: 0em;"><span style="display: inline-block; position: relative; width: 9.224em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1008.81em 4.306em -999.997em); top: -7.067em; left: 0em;"><span class="mtd" id="MathJax-Span-3452"><span class="mrow" id="MathJax-Span-3453"><span class="mi" id="MathJax-Span-3454" style="font-family: STIXGeneral;">ln</span><span class="mo" id="MathJax-Span-3455"></span><span class="mo" id="MathJax-Span-3456" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-3457" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mo" id="MathJax-Span-3458" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-3459" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-3460" style="font-family: STIXGeneral; padding-left: 0.259em;">ln</span><span class="mo" id="MathJax-Span-3461"></span><span class="mo" id="MathJax-Span-3462" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-3463" style="font-family: STIXGeneral; font-style: italic;">b</span><span class="mo" id="MathJax-Span-3464" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-3465" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-3466" style="font-family: STIXGeneral; padding-left: 0.31em;">ln</span><span class="mo" id="MathJax-Span-3467"></span><span class="mo" id="MathJax-Span-3468" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-3469" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-3470" style="font-family: STIXGeneral; font-style: italic;">b</span><span class="mo" id="MathJax-Span-3471" style="font-family: STIXGeneral;">)</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.666em 1009.12em 4.869em -999.997em); top: -5.377em; left: 0em;"><span class="mtd" id="MathJax-Span-3474"><span class="mrow" id="MathJax-Span-3475"><span class="mi" id="MathJax-Span-3476" style="font-family: STIXGeneral;">ln</span><span class="mo" id="MathJax-Span-3477"></span><span class="mo" id="MathJax-Span-3478" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-3479" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mo" id="MathJax-Span-3480" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-3481" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-3482" style="font-family: STIXGeneral; padding-left: 0.259em;">ln</span><span class="mo" id="MathJax-Span-3483"></span><span class="mo" id="MathJax-Span-3484" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-3485" style="font-family: STIXGeneral; font-style: italic;">b</span><span class="mo" id="MathJax-Span-3486" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-3487" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-3488" style="font-family: STIXGeneral; padding-left: 0.31em;">ln</span><span class="mo" id="MathJax-Span-3489"></span><span class="mrow" id="MathJax-Span-3490"><span class="mo" id="MathJax-Span-3491" style="vertical-align: -0.407em;"><span style="font-family: STIXSizeTwoSym;">(</span></span><span class="mfrac" id="MathJax-Span-3492"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-3493" style="font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.46em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-3494" style="font-family: STIXGeneral; font-style: italic;">b</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.62em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.617em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-3495" style="vertical-align: -0.407em;"><span style="font-family: STIXSizeTwoSym;">)</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.128em 1006.3em 4.306em -999.997em); top: -3.532em; left: 0em;"><span class="mtd" id="MathJax-Span-3498"><span class="mrow" id="MathJax-Span-3499"><span class="mi" id="MathJax-Span-3500" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mi" id="MathJax-Span-3501" style="font-family: STIXGeneral; padding-left: 0.207em;">ln</span><span class="mo" id="MathJax-Span-3502"></span><span class="mo" id="MathJax-Span-3503" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-3504" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mo" id="MathJax-Span-3505" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-3506" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-3507" style="font-family: STIXGeneral; padding-left: 0.31em;">ln</span><span class="mo" id="MathJax-Span-3508"></span><span class="mo" id="MathJax-Span-3509" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-3510"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-3511" style="font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.515em;"><span class="mi" id="MathJax-Span-3512" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-3513" style="font-family: STIXGeneral;">)</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.41em 1006.92em 5.074em -999.997em); top: -1.586em; left: 0em;"><span class="mtd" id="MathJax-Span-3516"><span class="mrow" id="MathJax-Span-3517"><span class="msubsup" id="MathJax-Span-3518"><span style="display: inline-block; position: relative; width: 1.693em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1001.23em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-3519" style="font-family: STIXGeneral;">log</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.737em; left: 1.283em;"><span class="mi" id="MathJax-Span-3520" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-3521"></span><span class="mi" id="MathJax-Span-3522" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">b</span><span class="mo" id="MathJax-Span-3523" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-3524" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 2.974em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1002.77em 4.408em -999.997em); top: -4.711em; left: 50%; margin-left: -1.432em;"><span class="mrow" id="MathJax-Span-3525"><span class="msubsup" id="MathJax-Span-3526"><span style="display: inline-block; position: relative; width: 1.693em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1001.23em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-3527" style="font-family: STIXGeneral;">log</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.737em; left: 1.283em;"><span class="mi" id="MathJax-Span-3528" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">c</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-3529"></span><span class="mo" id="MathJax-Span-3530" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-3531" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mo" id="MathJax-Span-3532" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1002.77em 4.408em -999.997em); top: -3.327em; left: 50%; margin-left: -1.432em;"><span class="mrow" id="MathJax-Span-3533"><span class="msubsup" id="MathJax-Span-3534"><span style="display: inline-block; position: relative; width: 1.693em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1001.23em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-3535" style="font-family: STIXGeneral;">log</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.737em; left: 1.283em;"><span class="mi" id="MathJax-Span-3536" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">c</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-3537"></span><span class="mo" id="MathJax-Span-3538" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-3539" style="font-family: STIXGeneral; font-style: italic;">b</span><span class="mo" id="MathJax-Span-3540" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1002.97em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.974em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 8.148em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -4.184em; border-left: 0px solid; width: 0px; height: 8.941em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd></mtd><mtd><mi>ln</mi><mo></mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>+</mo><mi>ln</mi><mo></mo><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ln</mi><mo></mo><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi>ln</mi><mo></mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>−</mo><mi>ln</mi><mo></mo><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ln</mi><mo></mo><mrow><mo>(</mo><mfrac><mi>a</mi><mi>b</mi></mfrac><mo>)</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mi>n</mi><mi>ln</mi><mo></mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ln</mi><mo></mo><mo stretchy="false">(</mo><msup><mi>a</mi><mi>n</mi></msup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><msub><mi>log</mi><mi>a</mi></msub><mo></mo><mi>b</mi><mo>=</mo><mfrac><mrow><msub><mi>log</mi><mi>c</mi></msub><mo></mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi>log</mi><mi>c</mi></msub><mo></mo><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mfrac></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-121">\begin{align}
&\ln(a) + \ln(b) = \ln(ab) \\
&\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \\
&n\ln(a) = \ln(a^n)\\
&\log_a b = \frac{\log_c(a)}{\log_c(b)}
\end{align}</script></span></p><h3 id="differentials-and-integrals"><a class="anchor hidden-xs" href="#differentials-and-integrals" title="differentials-and-integrals"><span class="octicon octicon-link"></span></a>Differentials and integrals</h3><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-129-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em 0em 0em" displaystyle="true"><mtr><mtd /><mtd><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><msup><mi>e</mi><mi>x</mi></msup><mo>=</mo><msup><mi>e</mi><mi>x</mi></msup><mspace width="1em" /></mtd><mtd /><mtd><mi></mi><mo>&#x222B;</mo><msup><mi>e</mi><mi>x</mi></msup><mi>d</mi><mi>x</mi><mo>=</mo><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><mi>C</mi></mtd></mtr><mtr><mtd /><mtd><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow></msup><mo>=</mo><mi>a</mi><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow></msup><mspace width="1em" /></mtd><mtd /><mtd><mi></mi><mo>&#x222B;</mo><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow></msup><mi>d</mi><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mi>a</mi></mfrac><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow></msup><mo>+</mo><mi>C</mi></mtd></mtr><mtr><mtd /><mtd><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msup><mo>=</mo><msup><mi>f</mi><mo>&#x2032;</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msup><mspace width="1em" /></mtd><mtd /><mtd><mi></mi><mo>&#x222B;</mo><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msup><mi>d</mi><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mi>f</mi><mo>&#x2032;</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msup><mo>+</mo><mi>C</mi></mtd></mtr><mtr><mtd /><mtd><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mi>ln</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><mspace width="1em" /></mtd><mtd /><mtd><mi></mi><mo>&#x222B;</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><mi>d</mi><mi>x</mi><mo>=</mo><mi>ln</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi>C</mi></mtd></mtr><mtr><mtd /><mtd><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mi>sin</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>c</mi><mi>o</mi><mi>s</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="1em" /></mtd><mtd /><mtd><mi></mi><mo>&#x222B;</mo><mi>sin</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>d</mi><mi>x</mi><mo>=</mo><mo>&#x2212;</mo><mi>cos</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi>C</mi></mtd></mtr><mtr><mtd /><mtd><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mi>c</mi><mi>o</mi><mi>s</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>&#x2212;</mo><mi>sin</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="1em" /></mtd><mtd /><mtd><mi></mi><mo>&#x222B;</mo><mi>cos</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>=</mo><mi>sin</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi>C</mi></mtd></mtr></mtable></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-5605" style="width: 27.154em; 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top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-5719" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.515em;"><span class="mo" id="MathJax-Span-5720" style="font-size: 70.7%; font-family: STIXVariants;">′</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-5721" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-5722" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-5723" style="font-family: STIXGeneral;">)</span><span class="msubsup" id="MathJax-Span-5724"><span style="display: inline-block; position: relative; width: 1.591em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-5725" style="font-family: STIXGeneral; font-style: italic;">e</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.464em;"><span class="texatom" id="MathJax-Span-5726"><span class="mrow" id="MathJax-Span-5727"><span class="mi" id="MathJax-Span-5728" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mo" id="MathJax-Span-5729" style="font-size: 70.7%; font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-5730" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-5731" style="font-size: 70.7%; font-family: STIXGeneral;">)</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mspace" id="MathJax-Span-5732" style="height: 0em; vertical-align: 0em; width: 1.13em; display: inline-block; overflow: hidden;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.462em 1006.51em 4.869em -999.997em); top: -2.661em; left: 0em;"><span class="mtd" id="MathJax-Span-5771"><span class="mrow" id="MathJax-Span-5772"><span class="mfrac" id="MathJax-Span-5773"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-5774" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.51em;"><span class="mrow" id="MathJax-Span-5775"><span class="mi" id="MathJax-Span-5776" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-5777" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.08em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.078em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mi" id="MathJax-Span-5778" style="font-family: STIXGeneral;">ln</span><span class="mo" id="MathJax-Span-5779"></span><span class="mo" id="MathJax-Span-5780" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-5781" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-5782" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-5783" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-5784" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mn" id="MathJax-Span-5785" style="font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.202em;"><span class="mi" id="MathJax-Span-5786" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.62em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.617em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mspace" id="MathJax-Span-5787" style="height: 0em; vertical-align: 0em; width: 1.13em; display: inline-block; overflow: hidden;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.462em 1008.51em 4.869em -999.997em); top: -0.1em; left: 0em;"><span class="mtd" id="MathJax-Span-5809"><span class="mrow" id="MathJax-Span-5810"><span class="mfrac" id="MathJax-Span-5811"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-5812" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.51em;"><span class="mrow" id="MathJax-Span-5813"><span class="mi" id="MathJax-Span-5814" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-5815" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.08em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.078em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mi" id="MathJax-Span-5816" style="font-family: STIXGeneral;">sin</span><span class="mo" id="MathJax-Span-5817"></span><span class="mo" id="MathJax-Span-5818" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-5819" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-5820" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-5821" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-5822" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">c</span><span class="mi" id="MathJax-Span-5823" style="font-family: STIXGeneral; font-style: italic;">o</span><span class="mi" id="MathJax-Span-5824" style="font-family: STIXGeneral; font-style: italic;">s</span><span class="mo" id="MathJax-Span-5825" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-5826" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-5827" style="font-family: STIXGeneral;">)</span><span class="mspace" id="MathJax-Span-5828" style="height: 0em; vertical-align: 0em; width: 1.13em; display: inline-block; overflow: hidden;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.462em 1009.38em 4.869em -999.997em); top: 2.513em; left: 0em;"><span class="mtd" id="MathJax-Span-5854"><span class="mrow" id="MathJax-Span-5855"><span class="mfrac" id="MathJax-Span-5856"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-5857" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.51em;"><span class="mrow" id="MathJax-Span-5858"><span class="mi" id="MathJax-Span-5859" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-5860" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.08em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.078em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mi" id="MathJax-Span-5861" style="font-family: STIXGeneral; font-style: italic;">c</span><span class="mi" id="MathJax-Span-5862" style="font-family: STIXGeneral; font-style: italic;">o</span><span class="mi" id="MathJax-Span-5863" style="font-family: STIXGeneral; font-style: italic;">s</span><span class="mo" id="MathJax-Span-5864" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-5865" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-5866" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-5867" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mo" id="MathJax-Span-5868" style="font-family: STIXGeneral; padding-left: 0.31em;">−</span><span class="mi" id="MathJax-Span-5869" style="font-family: STIXGeneral; padding-left: 0.207em;">sin</span><span class="mo" id="MathJax-Span-5870"></span><span class="mo" id="MathJax-Span-5871" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-5872" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-5873" style="font-family: STIXGeneral;">)</span><span class="mspace" id="MathJax-Span-5874" style="height: 0em; vertical-align: 0em; width: 1.13em; display: inline-block; overflow: hidden;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 16.089em;"></span></span><span style="position: absolute; clip: rect(7.38em 1000em 20.699em -999.997em); top: -14.034em; left: 9.378em;"><span style="display: inline-block; position: relative; width: 0em; height: 0px;"><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -10.499em; right: 0em;"><span class="mtd" id="MathJax-Span-5625"><span class="mrow" id="MathJax-Span-5626"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -7.887em; right: 0em;"><span class="mtd" id="MathJax-Span-5670"><span class="mrow" id="MathJax-Span-5671"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -5.325em; right: 0em;"><span class="mtd" id="MathJax-Span-5733"><span class="mrow" id="MathJax-Span-5734"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -2.661em; right: 0em;"><span class="mtd" id="MathJax-Span-5788"><span class="mrow" id="MathJax-Span-5789"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -0.1em; right: 0em;"><span class="mtd" id="MathJax-Span-5829"><span class="mrow" id="MathJax-Span-5830"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: 2.513em; right: 0em;"><span class="mtd" id="MathJax-Span-5875"><span class="mrow" id="MathJax-Span-5876"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 14.039em;"></span></span><span style="position: absolute; clip: rect(8.251em 1012.55em 23.876em -999.997em); top: -16.288em; left: 9.378em;"><span style="display: inline-block; position: relative; width: 12.554em; height: 0px;"><span style="position: absolute; clip: rect(2.462em 1007.28em 5.023em -999.997em); top: -10.499em; left: 0em;"><span class="mtd" id="MathJax-Span-5627"><span class="mrow" id="MathJax-Span-5628"><span class="mi" id="MathJax-Span-5629"></span><span class="mo" id="MathJax-Span-5630" style="font-family: STIXIntegralsD; vertical-align: -0.612em; padding-left: 0.207em;">∫<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.464em;"></span></span><span class="msubsup" id="MathJax-Span-5631" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 0.822em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-5632" style="font-family: STIXGeneral; font-style: italic;">e</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.464em;"><span class="mi" id="MathJax-Span-5633" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mi" id="MathJax-Span-5634" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-5635" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-5636" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="msubsup" id="MathJax-Span-5637" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.822em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-5638" style="font-family: STIXGeneral; font-style: italic;">e</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.464em;"><span class="mi" id="MathJax-Span-5639" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-5640" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-5641" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.462em 1010.56em 5.023em -999.997em); top: -7.887em; left: 0em;"><span class="mtd" id="MathJax-Span-5672"><span class="mrow" id="MathJax-Span-5673"><span class="mi" id="MathJax-Span-5674"></span><span class="mo" id="MathJax-Span-5675" style="font-family: STIXIntegralsD; vertical-align: -0.612em; padding-left: 0.207em;">∫<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.464em;"></span></span><span class="msubsup" id="MathJax-Span-5676" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 2.052em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-5677" style="font-family: STIXGeneral; font-style: italic;">e</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.464em;"><span class="texatom" id="MathJax-Span-5678"><span class="mrow" id="MathJax-Span-5679"><span class="mi" id="MathJax-Span-5680" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-5681" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-5682" style="font-size: 70.7%; font-family: STIXGeneral;">+</span><span class="mi" id="MathJax-Span-5683" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">b</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mi" id="MathJax-Span-5684" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-5685" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-5686" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-5687" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mn" id="MathJax-Span-5688" style="font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-5689" style="font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.62em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.617em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-5690"><span style="display: inline-block; position: relative; width: 2.052em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-5691" style="font-family: STIXGeneral; font-style: italic;">e</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.464em;"><span class="texatom" id="MathJax-Span-5692"><span class="mrow" id="MathJax-Span-5693"><span class="mi" id="MathJax-Span-5694" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-5695" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-5696" style="font-size: 70.7%; font-family: STIXGeneral;">+</span><span class="mi" id="MathJax-Span-5697" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">b</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-5698" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-5699" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.462em 1011.12em 5.023em -999.997em); top: -5.325em; left: 0em;"><span class="mtd" id="MathJax-Span-5735"><span class="mrow" id="MathJax-Span-5736"><span class="mi" id="MathJax-Span-5737"></span><span class="mo" id="MathJax-Span-5738" style="font-family: STIXIntegralsD; vertical-align: -0.612em; padding-left: 0.207em;">∫<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.464em;"></span></span><span class="msubsup" id="MathJax-Span-5739" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 1.591em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-5740" style="font-family: STIXGeneral; font-style: italic;">e</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.464em;"><span class="texatom" id="MathJax-Span-5741"><span class="mrow" id="MathJax-Span-5742"><span class="mi" id="MathJax-Span-5743" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mo" id="MathJax-Span-5744" style="font-size: 70.7%; font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-5745" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-5746" style="font-size: 70.7%; font-family: STIXGeneral;">)</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mi" id="MathJax-Span-5747" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-5748" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-5749" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-5750" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 2.052em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mn" id="MathJax-Span-5751" style="font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1001.9em 4.357em -999.997em); top: -3.327em; left: 50%; margin-left: -0.971em;"><span class="mrow" id="MathJax-Span-5752"><span class="msup" id="MathJax-Span-5753"><span style="display: inline-block; position: relative; width: 0.822em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-5754" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.301em; left: 0.515em;"><span class="mo" id="MathJax-Span-5755" style="font-size: 70.7%; font-family: STIXVariants;">′</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-5756" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-5757" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-5758" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1002.05em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.052em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-5759"><span style="display: inline-block; position: relative; width: 1.591em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-5760" style="font-family: STIXGeneral; font-style: italic;">e</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.464em;"><span class="texatom" id="MathJax-Span-5761"><span class="mrow" id="MathJax-Span-5762"><span class="mi" id="MathJax-Span-5763" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mo" id="MathJax-Span-5764" style="font-size: 70.7%; font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-5765" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-5766" style="font-size: 70.7%; font-family: STIXGeneral;">)</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-5767" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-5768" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.462em 1008.35em 5.023em -999.997em); top: -2.661em; left: 0em;"><span class="mtd" id="MathJax-Span-5790"><span class="mrow" id="MathJax-Span-5791"><span class="mi" id="MathJax-Span-5792"></span><span class="mo" id="MathJax-Span-5793" style="font-family: STIXIntegralsD; vertical-align: -0.612em; padding-left: 0.207em;">∫<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.464em;"></span></span><span class="mfrac" id="MathJax-Span-5794" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mn" id="MathJax-Span-5795" style="font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.202em;"><span class="mi" id="MathJax-Span-5796" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.62em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.617em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mi" id="MathJax-Span-5797" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-5798" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-5799" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-5800" style="font-family: STIXGeneral; padding-left: 0.31em;">ln</span><span class="mo" id="MathJax-Span-5801"></span><span class="mo" id="MathJax-Span-5802" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-5803" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-5804" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-5805" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-5806" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.462em 1012.55em 5.023em -999.997em); top: -0.1em; left: 0em;"><span class="mtd" id="MathJax-Span-5831"><span class="mrow" id="MathJax-Span-5832"><span class="mi" id="MathJax-Span-5833"></span><span class="mo" id="MathJax-Span-5834" style="font-family: STIXIntegralsD; vertical-align: -0.612em; padding-left: 0.207em;">∫<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.464em;"></span></span><span class="mi" id="MathJax-Span-5835" style="font-family: STIXGeneral; padding-left: 0.207em;">sin</span><span class="mo" id="MathJax-Span-5836"></span><span class="mo" id="MathJax-Span-5837" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-5838" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-5839" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-5840" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-5841" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-5842" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-5843" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mo" id="MathJax-Span-5844" style="font-family: STIXGeneral; padding-left: 0.31em;">−</span><span class="mi" id="MathJax-Span-5845" style="font-family: STIXGeneral; padding-left: 0.207em;">cos</span><span class="mo" id="MathJax-Span-5846"></span><span class="mo" id="MathJax-Span-5847" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-5848" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-5849" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-5850" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-5851" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.462em 1010.35em 5.023em -999.997em); top: 2.513em; left: 0em;"><span class="mtd" id="MathJax-Span-5877"><span class="mrow" id="MathJax-Span-5878"><span class="mi" id="MathJax-Span-5879"></span><span class="mo" id="MathJax-Span-5880" style="font-family: STIXIntegralsD; vertical-align: -0.612em; padding-left: 0.207em;">∫<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.464em;"></span></span><span class="mi" id="MathJax-Span-5881" style="font-family: STIXGeneral; padding-left: 0.207em;">cos</span><span class="mo" id="MathJax-Span-5882"></span><span class="mo" id="MathJax-Span-5883" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-5884" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-5885" style="font-family: STIXGeneral;">)</span><span class="mi" id="MathJax-Span-5886" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-5887" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-5888" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-5889" style="font-family: STIXGeneral; padding-left: 0.31em;">sin</span><span class="mo" id="MathJax-Span-5890"></span><span class="mo" id="MathJax-Span-5891" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-5892" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-5893" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-5894" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-5895" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 16.294em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -9.122em; border-left: 0px solid; width: 0px; height: 18.816em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em 0em 0em" displaystyle="true"><mtr><mtd></mtd><mtd><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><msup><mi>e</mi><mi>x</mi></msup><mo>=</mo><msup><mi>e</mi><mi>x</mi></msup><mspace width="1em"></mspace></mtd><mtd></mtd><mtd><mi></mi><mo>∫</mo><msup><mi>e</mi><mi>x</mi></msup><mi>d</mi><mi>x</mi><mo>=</mo><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><mi>C</mi></mtd></mtr><mtr><mtd></mtd><mtd><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow></msup><mo>=</mo><mi>a</mi><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow></msup><mspace width="1em"></mspace></mtd><mtd></mtd><mtd><mi></mi><mo>∫</mo><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow></msup><mi>d</mi><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mi>a</mi></mfrac><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow></msup><mo>+</mo><mi>C</mi></mtd></mtr><mtr><mtd></mtd><mtd><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msup><mo>=</mo><msup><mi>f</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msup><mspace width="1em"></mspace></mtd><mtd></mtd><mtd><mi></mi><mo>∫</mo><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msup><mi>d</mi><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mi>f</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msup><mo>+</mo><mi>C</mi></mtd></mtr><mtr><mtd></mtd><mtd><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mi>ln</mi><mo></mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><mspace width="1em"></mspace></mtd><mtd></mtd><mtd><mi></mi><mo>∫</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><mi>d</mi><mi>x</mi><mo>=</mo><mi>ln</mi><mo></mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi>C</mi></mtd></mtr><mtr><mtd></mtd><mtd><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mi>sin</mi><mo></mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>c</mi><mi>o</mi><mi>s</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="1em"></mspace></mtd><mtd></mtd><mtd><mi></mi><mo>∫</mo><mi>sin</mi><mo></mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>d</mi><mi>x</mi><mo>=</mo><mo>−</mo><mi>cos</mi><mo></mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi>C</mi></mtd></mtr><mtr><mtd></mtd><mtd><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mi>c</mi><mi>o</mi><mi>s</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mi>sin</mi><mo></mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="1em"></mspace></mtd><mtd></mtd><mtd><mi></mi><mo>∫</mo><mi>cos</mi><mo></mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>=</mo><mi>sin</mi><mo></mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi>C</mi></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-129">\begin{alignat*}{3}
&\frac{d}{dx} e^x = e^x \quad &&\int e^x dx = e^x + C \\
&\frac{d}{dx} e^{ax + b} = ae^{ax+b} \quad &&\int e^{ax+b} dx = \frac{1}{a} e^{ax+b} + C\\
&\frac{d}{dx} e^{f(x)} = f'(x)e^{f(x)} \quad &&\int e^{f(x)} dx = \frac{1}{f'(x)} e^{f(x)} + C \\
&\frac{d}{dx} \ln(x) = \frac{1}{x} \quad &&\int \frac{1}{x} dx = \ln(x) + C \\
&\frac{d}{dx} \sin(x) = cos(x) \quad &&\int \sin(x) = dx = -\cos(x) + C \\
&\frac{d}{dx} cos(x) = -\sin(x) \quad &&\int \cos(x) dx = \sin(x) + C\end{alignat*}</script></span></p><h3 id="the-chain-rule"><a class="anchor hidden-xs" href="#the-chain-rule" title="the-chain-rule"><span class="octicon octicon-link"></span></a>The chain rule</h3><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-22-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mtext>&#xA0;</mtext><mo>=</mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>u</mi></mrow></mfrac><mfrac><mrow><mi>d</mi><mi>u</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-684" style="width: 6.765em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.535em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.027em 1005.53em 3.435em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-685"><span class="mfrac" id="MathJax-Span-686"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.357em -999.997em); top: -4.659em; left: 50%; margin-left: -0.459em;"><span class="mrow" id="MathJax-Span-687"><span class="mi" id="MathJax-Span-688" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-689" style="font-family: STIXGeneral; font-style: italic;">y</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.51em;"><span class="mrow" id="MathJax-Span-690"><span class="mi" id="MathJax-Span-691" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-692" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.08em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.078em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mtext" id="MathJax-Span-693" style="font-family: STIXGeneral;"> </span><span class="mo" id="MathJax-Span-694" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-695" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.357em -999.997em); top: -4.659em; left: 50%; margin-left: -0.459em;"><span class="mrow" id="MathJax-Span-696"><span class="mi" id="MathJax-Span-697" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-698" style="font-family: STIXGeneral; font-style: italic;">y</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1001.03em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.51em;"><span class="mrow" id="MathJax-Span-699"><span class="mi" id="MathJax-Span-700" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-701" style="font-family: STIXGeneral; font-style: italic;">u</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.13em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.13em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mfrac" id="MathJax-Span-702"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1001.03em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.51em;"><span class="mrow" id="MathJax-Span-703"><span class="mi" id="MathJax-Span-704" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-705" style="font-family: STIXGeneral; font-style: italic;">u</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.51em;"><span class="mrow" id="MathJax-Span-706"><span class="mi" id="MathJax-Span-707" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-708" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.13em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.13em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.934em; border-left: 0px solid; width: 0px; height: 2.628em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mtext> </mtext><mo>=</mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>u</mi></mrow></mfrac><mfrac><mrow><mi>d</mi><mi>u</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-22">\frac{dy}{dx}\ = \frac{dy}{du} \frac{du}{dx}</script></span></p><h3 id="the-product-rule"><a class="anchor hidden-xs" href="#the-product-rule" title="the-product-rule"><span class="octicon octicon-link"></span></a>The product rule</h3><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-23-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>g</mi><mo>&#x2032;</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mi>f</mi><mo>&#x2032;</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-709" style="width: 17.011em; display: inline-block;"><span style="display: inline-block; position: relative; width: 13.937em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.027em 1013.89em 3.435em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-710"><span class="mfrac" id="MathJax-Span-711"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-712" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.51em;"><span class="mrow" id="MathJax-Span-713"><span class="mi" id="MathJax-Span-714" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-715" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.08em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.078em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mi" id="MathJax-Span-716" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-717" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-718" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-719" style="font-family: STIXGeneral;">)</span><span class="mi" id="MathJax-Span-720" style="font-family: STIXGeneral; font-style: italic;">g</span><span class="mo" id="MathJax-Span-721" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-722" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-723" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-724" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-725" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-726" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-727" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-728" style="font-family: STIXGeneral;">)</span><span class="msup" id="MathJax-Span-729"><span style="display: inline-block; position: relative; width: 0.822em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-730" style="font-family: STIXGeneral; font-style: italic;">g</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.515em;"><span class="mo" id="MathJax-Span-731" style="font-size: 70.7%; font-family: STIXVariants;">′</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-732" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-733" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-734" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-735" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="msup" id="MathJax-Span-736" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.822em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-737" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.515em;"><span class="mo" id="MathJax-Span-738" style="font-size: 70.7%; font-family: STIXVariants;">′</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-739" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-740" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-741" style="font-family: STIXGeneral;">)</span><span class="mi" id="MathJax-Span-742" style="font-family: STIXGeneral; font-style: italic;">g</span><span class="mo" id="MathJax-Span-743" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-744" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-745" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.934em; border-left: 0px solid; width: 0px; height: 2.628em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>g</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mi>f</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-23"> \frac{d}{dx} f(x) g(x) = f(x)g'(x) + f'(x)g(x) </script></span></p><p>or</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-24-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mi>u</mi><mi>v</mi><mo>=</mo><mi>u</mi><msup><mi>v</mi><mo>&#x2032;</mo></msup><mo>+</mo><mi>v</mi><msup><mi>u</mi><mo>&#x2032;</mo></msup></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-746" style="width: 8.916em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.277em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.027em 1007.28em 3.435em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-747"><span class="mfrac" id="MathJax-Span-748"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-749" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.51em;"><span class="mrow" id="MathJax-Span-750"><span class="mi" id="MathJax-Span-751" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-752" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.08em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.078em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mi" id="MathJax-Span-753" style="font-family: STIXGeneral; font-style: italic;">u</span><span class="mi" id="MathJax-Span-754" style="font-family: STIXGeneral; font-style: italic;">v</span><span class="mo" id="MathJax-Span-755" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-756" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">u</span><span class="msup" id="MathJax-Span-757"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-758" style="font-family: STIXGeneral; font-style: italic;">v</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.464em;"><span class="mo" id="MathJax-Span-759" style="font-size: 70.7%; font-family: STIXVariants;">′</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-760" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-761" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">v</span><span class="msup" id="MathJax-Span-762"><span style="display: inline-block; position: relative; width: 0.822em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-763" style="font-family: STIXGeneral; font-style: italic;">u</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.515em;"><span class="mo" id="MathJax-Span-764" style="font-size: 70.7%; font-family: STIXVariants;">′</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.934em; border-left: 0px solid; width: 0px; height: 2.628em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mi>u</mi><mi>v</mi><mo>=</mo><mi>u</mi><msup><mi>v</mi><mo>′</mo></msup><mo>+</mo><mi>v</mi><msup><mi>u</mi><mo>′</mo></msup></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-24">\frac{d}{dx} uv = uv' + vu'</script></span></p><h3 id="the-quotient-rule"><a class="anchor hidden-xs" href="#the-quotient-rule" title="the-quotient-rule"><span class="octicon octicon-link"></span></a>The quotient rule</h3><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-25-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mfrac><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>f</mi><mo>&#x2032;</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>&#x2212;</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>g</mi><mo>&#x2032;</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-765" style="width: 16.037em; display: inline-block;"><span style="display: inline-block; position: relative; width: 13.117em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.976em 1013.12em 3.64em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-766"><span class="mfrac" id="MathJax-Span-767"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-768" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.51em;"><span class="mrow" id="MathJax-Span-769"><span class="mi" id="MathJax-Span-770" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-771" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.08em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.078em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mfrac" id="MathJax-Span-772"><span style="display: inline-block; position: relative; width: 1.744em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1001.49em 4.357em -999.997em); top: -4.659em; left: 50%; margin-left: -0.766em;"><span class="mrow" id="MathJax-Span-773"><span class="mi" id="MathJax-Span-774" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-775" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-776" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-777" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1001.59em 4.357em -999.997em); top: -3.327em; left: 50%; margin-left: -0.817em;"><span class="mrow" id="MathJax-Span-778"><span class="mi" id="MathJax-Span-779" style="font-family: STIXGeneral; font-style: italic;">g</span><span class="mo" id="MathJax-Span-780" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-781" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-782" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.74em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.744em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-783" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-784" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 8.353em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.076em 1008.2em 4.357em -999.997em); top: -4.659em; left: 50%; margin-left: -4.096em;"><span class="mrow" id="MathJax-Span-785"><span class="mi" id="MathJax-Span-786" style="font-family: STIXGeneral; font-style: italic;">g</span><span class="mo" id="MathJax-Span-787" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-788" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-789" style="font-family: STIXGeneral;">)</span><span class="msup" id="MathJax-Span-790"><span style="display: inline-block; position: relative; width: 0.822em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-791" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.515em;"><span class="mo" id="MathJax-Span-792" style="font-size: 70.7%; font-family: STIXVariants;">′</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-793" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-794" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-795" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-796" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-797" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-798" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-799" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-800" style="font-family: STIXGeneral;">)</span><span class="msup" id="MathJax-Span-801"><span style="display: inline-block; position: relative; width: 0.822em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-802" style="font-family: STIXGeneral; font-style: italic;">g</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.515em;"><span class="mo" id="MathJax-Span-803" style="font-size: 70.7%; font-family: STIXVariants;">′</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-804" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-805" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-806" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.076em 1002.05em 4.357em -999.997em); top: -3.276em; left: 50%; margin-left: -1.022em;"><span class="mrow" id="MathJax-Span-807"><span class="mi" id="MathJax-Span-808" style="font-family: STIXGeneral; font-style: italic;">g</span><span class="mo" id="MathJax-Span-809" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-810" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-811"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.31em 4.306em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-812" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.301em; left: 0.361em;"><span class="mn" id="MathJax-Span-813" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1008.35em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 8.353em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.184em; border-left: 0px solid; width: 0px; height: 3.003em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mfrac><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>f</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>g</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-25">\frac{d}{dx} \frac{f(x)}{g(x)} = \frac{g(x)f'(x) - f(x)g'(x)}{g(x)^2}</script></span></p><p>or</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-26-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mfrac><mi>u</mi><mi>v</mi></mfrac><mo>=</mo><mfrac><mrow><mi>v</mi><msup><mi>u</mi><mo>&#x2032;</mo></msup><mo>&#x2212;</mo><mi>u</mi><msup><mi>v</mi><mo>&#x2032;</mo></msup></mrow><msup><mi>v</mi><mn>2</mn></msup></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-814" style="width: 9.121em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.482em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.976em 1007.48em 3.486em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-815"><span class="mfrac" id="MathJax-Span-816"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-817" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.51em;"><span class="mrow" id="MathJax-Span-818"><span class="mi" id="MathJax-Span-819" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-820" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.08em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.078em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mfrac" id="MathJax-Span-821"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-822" style="font-family: STIXGeneral; font-style: italic;">u</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.202em;"><span class="mi" id="MathJax-Span-823" style="font-family: STIXGeneral; font-style: italic;">v</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.62em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.617em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-824" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-825" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 3.845em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.076em 1003.74em 4.306em -999.997em); top: -4.659em; left: 50%; margin-left: -1.842em;"><span class="mrow" id="MathJax-Span-826"><span class="mi" id="MathJax-Span-827" style="font-family: STIXGeneral; font-style: italic;">v</span><span class="msup" id="MathJax-Span-828"><span style="display: inline-block; position: relative; width: 0.822em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-829" style="font-family: STIXGeneral; font-style: italic;">u</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.515em;"><span class="mo" id="MathJax-Span-830" style="font-size: 70.7%; font-family: STIXVariants;">′</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-831" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-832" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">u</span><span class="msup" id="MathJax-Span-833"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-834" style="font-family: STIXGeneral; font-style: italic;">v</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.464em;"><span class="mo" id="MathJax-Span-835" style="font-size: 70.7%; font-family: STIXVariants;">′</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.076em 1000.87em 4.152em -999.997em); top: -3.276em; left: 50%; margin-left: -0.459em;"><span class="msubsup" id="MathJax-Span-836"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-837" style="font-family: STIXGeneral; font-style: italic;">v</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.301em; left: 0.464em;"><span class="mn" id="MathJax-Span-838" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1003.85em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.845em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.997em; border-left: 0px solid; width: 0px; height: 2.816em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mfrac><mi>u</mi><mi>v</mi></mfrac><mo>=</mo><mfrac><mrow><mi>v</mi><msup><mi>u</mi><mo>′</mo></msup><mo>−</mo><mi>u</mi><msup><mi>v</mi><mo>′</mo></msup></mrow><msup><mi>v</mi><mn>2</mn></msup></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-26">\frac{d}{dx} \frac{u}{v} = \frac{vu' - uv'}{v^2}</script></span></p><h3 id="integration-by-inspection-and-substitution"><a class="anchor hidden-xs" href="#integration-by-inspection-and-substitution" title="integration-by-inspection-and-substitution"><span class="octicon octicon-link"></span></a>Integration by inspection and substitution</h3><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-135-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd /><mtd><mi></mi><mo>&#x222B;</mo><mi>cos</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mi>a</mi></mfrac><mi>sin</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo>+</mo><mi>C</mi></mtd></mtr><mtr><mtd /><mtd><mi></mi><mo>&#x222B;</mo><mi>sin</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>=</mo><mo>&#x2212;</mo><mfrac><mn>1</mn><mi>a</mi></mfrac><mi>cos</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo>+</mo><mi>C</mi></mtd></mtr><mtr><mtd /><mtd><mi></mi><mo>&#x222B;</mo><mi>sec</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mi>d</mi><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mi>a</mi></mfrac><mi>tan</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo>+</mo><mi>C</mi></mtd></mtr><mtr><mtd /><mtd><mi></mi><mo>&#x222B;</mo><mfrac><mrow><msup><mi>f</mi><mo>&#x2032;</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mi>d</mi><mi>x</mi><mo>=</mo><mi>l</mi><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>+</mo><mi>C</mi></mtd></mtr></mtable></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-6571" style="width: 20.289em; display: inline-block;"><span style="display: inline-block; position: relative; width: 16.601em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(-3.276em 1016.45em 7.175em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-6572"><span class="mtable" id="MathJax-Span-6573" style="padding-right: 0.156em; padding-left: 0.156em;"><span style="display: inline-block; position: relative; width: 16.294em; height: 0px;"><span style="position: absolute; clip: rect(4.767em 1000em 12.964em -999.997em); top: -8.86em; left: 0em;"><span style="display: inline-block; position: relative; width: 0em; height: 0px;"><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -7.938em; right: 0em;"><span class="mtd" id="MathJax-Span-6574"><span class="mrow" id="MathJax-Span-6575"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -5.325em; right: 0em;"><span class="mtd" id="MathJax-Span-6604"><span class="mrow" id="MathJax-Span-6605"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -2.713em; right: 0em;"><span class="mtd" id="MathJax-Span-6635"><span class="mrow" id="MathJax-Span-6636"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -0.049em; right: 0em;"><span class="mtd" id="MathJax-Span-6667"><span class="mrow" id="MathJax-Span-6668"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 8.865em;"></span></span><span style="position: absolute; clip: rect(5.689em 1016.29em 16.14em -999.997em); top: -11.165em; left: 0em;"><span style="display: inline-block; position: relative; width: 16.294em; height: 0px;"><span style="position: absolute; clip: rect(2.462em 1015.63em 5.023em -999.997em); top: -7.938em; left: 0em;"><span class="mtd" id="MathJax-Span-6576"><span class="mrow" id="MathJax-Span-6577"><span class="mi" id="MathJax-Span-6578"></span><span class="mo" id="MathJax-Span-6579" style="font-family: STIXIntegralsD; vertical-align: -0.612em; padding-left: 0.207em;">∫<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.464em;"></span></span><span class="mi" id="MathJax-Span-6580" style="font-family: STIXGeneral; padding-left: 0.207em;">cos</span><span class="mo" id="MathJax-Span-6581"></span><span class="mo" id="MathJax-Span-6582" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-6583" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-6584" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-6585" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-6586" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">b</span><span class="mo" id="MathJax-Span-6587" style="font-family: STIXGeneral;">)</span><span class="mi" id="MathJax-Span-6588" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-6589" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-6590" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-6591" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mn" id="MathJax-Span-6592" style="font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-6593" style="font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.62em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.617em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mi" id="MathJax-Span-6594" style="font-family: STIXGeneral;">sin</span><span class="mo" id="MathJax-Span-6595"></span><span class="mo" id="MathJax-Span-6596" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-6597" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-6598" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-6599" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-6600" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">b</span><span class="mo" id="MathJax-Span-6601" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-6602" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-6603" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.462em 1016.29em 5.023em -999.997em); top: -5.325em; left: 0em;"><span class="mtd" id="MathJax-Span-6606"><span class="mrow" id="MathJax-Span-6607"><span class="mi" id="MathJax-Span-6608"></span><span class="mo" id="MathJax-Span-6609" style="font-family: STIXIntegralsD; vertical-align: -0.612em; padding-left: 0.207em;">∫<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.464em;"></span></span><span class="mi" id="MathJax-Span-6610" style="font-family: STIXGeneral; padding-left: 0.207em;">sin</span><span class="mo" id="MathJax-Span-6611"></span><span class="mo" id="MathJax-Span-6612" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-6613" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-6614" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-6615" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-6616" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">b</span><span class="mo" id="MathJax-Span-6617" style="font-family: STIXGeneral;">)</span><span class="mi" id="MathJax-Span-6618" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-6619" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-6620" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mo" id="MathJax-Span-6621" style="font-family: STIXGeneral; padding-left: 0.31em;">−</span><span class="mfrac" id="MathJax-Span-6622"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mn" id="MathJax-Span-6623" style="font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-6624" style="font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.62em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.617em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mi" id="MathJax-Span-6625" style="font-family: STIXGeneral;">cos</span><span class="mo" id="MathJax-Span-6626"></span><span class="mo" id="MathJax-Span-6627" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-6628" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-6629" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-6630" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-6631" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">b</span><span class="mo" id="MathJax-Span-6632" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-6633" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-6634" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.462em 1016.04em 5.023em -999.997em); top: -2.713em; left: 0em;"><span class="mtd" id="MathJax-Span-6637"><span class="mrow" id="MathJax-Span-6638"><span class="mi" id="MathJax-Span-6639"></span><span class="mo" id="MathJax-Span-6640" style="font-family: STIXIntegralsD; vertical-align: -0.612em; padding-left: 0.207em;">∫<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.464em;"></span></span><span class="mi" id="MathJax-Span-6641" style="font-family: STIXGeneral; padding-left: 0.207em;">sec</span><span class="mo" id="MathJax-Span-6642"></span><span class="mo" id="MathJax-Span-6643" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-6644" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-6645" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-6646" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-6647" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">b</span><span class="msubsup" id="MathJax-Span-6648"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.31em 4.306em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-6649" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.361em;"><span class="mn" id="MathJax-Span-6650" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mi" id="MathJax-Span-6651" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-6652" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-6653" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-6654" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mn" id="MathJax-Span-6655" style="font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-6656" style="font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.62em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.617em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mi" id="MathJax-Span-6657" style="font-family: STIXGeneral;">tan</span><span class="mo" id="MathJax-Span-6658"></span><span class="mo" id="MathJax-Span-6659" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-6660" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-6661" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-6662" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-6663" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">b</span><span class="mo" id="MathJax-Span-6664" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-6665" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-6666" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.41em 1010.86em 5.023em -999.997em); top: -0.049em; left: 0em;"><span class="mtd" id="MathJax-Span-6669"><span class="mrow" id="MathJax-Span-6670"><span class="mi" id="MathJax-Span-6671"></span><span class="mo" id="MathJax-Span-6672" style="font-family: STIXIntegralsD; vertical-align: -0.612em; padding-left: 0.207em;">∫<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.464em;"></span></span><span class="mfrac" id="MathJax-Span-6673" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 2.052em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.076em 1001.9em 4.357em -999.997em); top: -4.659em; left: 50%; margin-left: -0.971em;"><span class="mrow" id="MathJax-Span-6674"><span class="msup" id="MathJax-Span-6675"><span style="display: inline-block; position: relative; width: 0.822em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-6676" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.515em;"><span class="mo" id="MathJax-Span-6677" style="font-size: 70.7%; font-family: STIXVariants;">′</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-6678" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-6679" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-6680" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1001.49em 4.357em -999.997em); top: -3.327em; left: 50%; margin-left: -0.766em;"><span class="mrow" id="MathJax-Span-6681"><span class="mi" id="MathJax-Span-6682" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-6683" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-6684" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-6685" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1002.05em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.052em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mi" id="MathJax-Span-6686" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-6687" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-6688" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-6689" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-6690" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="texatom" id="MathJax-Span-6691"><span class="mrow" id="MathJax-Span-6692"><span class="mo" id="MathJax-Span-6693" style="font-family: STIXVariants;">|</span></span></span><span class="mi" id="MathJax-Span-6694" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-6695" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-6696" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-6697" style="font-family: STIXGeneral;">)</span><span class="texatom" id="MathJax-Span-6698"><span class="mrow" id="MathJax-Span-6699"><span class="mo" id="MathJax-Span-6700" style="font-family: STIXVariants;">|</span></span></span><span class="mo" id="MathJax-Span-6701" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-6702" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 11.171em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -5.934em; border-left: 0px solid; width: 0px; height: 12.503em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd></mtd><mtd><mi></mi><mo>∫</mo><mi>cos</mi><mo></mo><mo stretchy="false">(</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mi>a</mi></mfrac><mi>sin</mi><mo></mo><mo stretchy="false">(</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo>+</mo><mi>C</mi></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>∫</mo><mi>sin</mi><mo></mo><mo stretchy="false">(</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mi>a</mi></mfrac><mi>cos</mi><mo></mo><mo stretchy="false">(</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo>+</mo><mi>C</mi></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>∫</mo><mi>sec</mi><mo></mo><mo stretchy="false">(</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mi>d</mi><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mi>a</mi></mfrac><mi>tan</mi><mo></mo><mo stretchy="false">(</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo>+</mo><mi>C</mi></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>∫</mo><mfrac><mrow><msup><mi>f</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mi>d</mi><mi>x</mi><mo>=</mo><mi>l</mi><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>+</mo><mi>C</mi></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-135">\begin{align} &\int \cos(ax + b) dx = \frac{1}{a} \sin(ax + b) + C \\
&\int \sin(ax + b) dx = -\frac{1}{a} \cos(ax + b) + C \\
&\int \sec(ax + b)^2 dx = \frac{1}{a}\tan(ax + b) + C \\
&\int \frac{f'(x)}{f(x)} dx = ln|f(x)| + C \end{align}</script></span></p><h3 id="integration-by-parts-and-standard-integrals"><a class="anchor hidden-xs" href="#integration-by-parts-and-standard-integrals" title="integration-by-parts-and-standard-integrals"><span class="octicon octicon-link"></span></a>Integration by parts and standard integrals</h3><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-28-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo>&#x222B;</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mtext>&#xA0;</mtext><mi>d</mi><mi>x</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>&#x2212;</mo><mo>&#x222B;</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mtext>&#xA0;</mtext><mi>d</mi><mi>x</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-977" style="width: 23.824em; display: inline-block;"><span style="display: inline-block; position: relative; width: 19.521em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(2.154em 1019.52em 4.716em -999.997em); top: -3.686em; left: 0em;"><span class="mrow" id="MathJax-Span-978"><span class="mo" id="MathJax-Span-979" style="font-family: STIXIntegralsD; vertical-align: -0.612em;">∫<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.464em;"></span></span><span class="mi" id="MathJax-Span-980" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-981" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-982" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-983" style="font-family: STIXGeneral;">)</span><span class="mfrac" id="MathJax-Span-984"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-985" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.51em;"><span class="mrow" id="MathJax-Span-986"><span class="mi" id="MathJax-Span-987" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-988" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.08em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.078em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mi" id="MathJax-Span-989" style="font-family: STIXGeneral; font-style: italic;">g</span><span class="mo" id="MathJax-Span-990" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-991" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-992" style="font-family: STIXGeneral;">)</span><span class="mtext" id="MathJax-Span-993" style="font-family: STIXGeneral;"> </span><span class="mi" id="MathJax-Span-994" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-995" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-996" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-997" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-998" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-999" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1000" style="font-family: STIXGeneral;">)</span><span class="mi" id="MathJax-Span-1001" style="font-family: STIXGeneral; font-style: italic;">g</span><span class="mo" id="MathJax-Span-1002" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1003" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1004" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1005" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mo" id="MathJax-Span-1006" style="font-family: STIXIntegralsD; vertical-align: -0.612em; padding-left: 0.259em;">∫<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.464em;"></span></span><span class="mi" id="MathJax-Span-1007" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">g</span><span class="mo" id="MathJax-Span-1008" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1009" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1010" style="font-family: STIXGeneral;">)</span><span class="mfrac" id="MathJax-Span-1011"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-1012" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.51em;"><span class="mrow" id="MathJax-Span-1013"><span class="mi" id="MathJax-Span-1014" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-1015" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.08em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.078em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mi" id="MathJax-Span-1016" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-1017" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1018" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1019" style="font-family: STIXGeneral;">)</span><span class="mtext" id="MathJax-Span-1020" style="font-family: STIXGeneral;"> </span><span class="mi" id="MathJax-Span-1021" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-1022" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.691em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.122em; border-left: 0px solid; width: 0px; height: 2.878em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo>∫</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mtext> </mtext><mi>d</mi><mi>x</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mo>∫</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mtext> </mtext><mi>d</mi><mi>x</mi></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-28">\int f(x) \frac{d}{dx} g(x)\ dx = f(x)g(x) - \int g(x) \frac{d}{dx} f(x) \ dx</script></span></p><p>More simply</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-29-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo>&#x222B;</mo><mi>u</mi><mfrac><mrow><mi>d</mi><mi>v</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mtext>&#xA0;</mtext><mi>d</mi><mi>x</mi><mo>=</mo><mi>u</mi><mi>v</mi><mo>&#x2212;</mo><mo>&#x222B;</mo><mi>v</mi><mfrac><mrow><mi>d</mi><mi>u</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mtext>&#xA0;</mtext><mi>d</mi><mi>x</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1023" style="width: 14.654em; display: inline-block;"><span style="display: inline-block; position: relative; width: 11.99em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(2.154em 1011.99em 4.716em -999.997em); top: -3.686em; left: 0em;"><span class="mrow" id="MathJax-Span-1024"><span class="mo" id="MathJax-Span-1025" style="font-family: STIXIntegralsD; vertical-align: -0.612em;">∫<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.464em;"></span></span><span class="mi" id="MathJax-Span-1026" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">u</span><span class="mfrac" id="MathJax-Span-1027"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.459em;"><span class="mrow" id="MathJax-Span-1028"><span class="mi" id="MathJax-Span-1029" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-1030" style="font-family: STIXGeneral; font-style: italic;">v</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.51em;"><span class="mrow" id="MathJax-Span-1031"><span class="mi" id="MathJax-Span-1032" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-1033" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.08em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.078em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mtext" id="MathJax-Span-1034" style="font-family: STIXGeneral;"> </span><span class="mi" id="MathJax-Span-1035" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-1036" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1037" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-1038" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">u</span><span class="mi" id="MathJax-Span-1039" style="font-family: STIXGeneral; font-style: italic;">v</span><span class="mo" id="MathJax-Span-1040" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mo" id="MathJax-Span-1041" style="font-family: STIXIntegralsD; vertical-align: -0.612em; padding-left: 0.259em;">∫<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.464em;"></span></span><span class="mi" id="MathJax-Span-1042" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">v</span><span class="mfrac" id="MathJax-Span-1043"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1001.03em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.51em;"><span class="mrow" id="MathJax-Span-1044"><span class="mi" id="MathJax-Span-1045" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-1046" style="font-family: STIXGeneral; font-style: italic;">u</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.51em;"><span class="mrow" id="MathJax-Span-1047"><span class="mi" id="MathJax-Span-1048" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-1049" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.13em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.13em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mtext" id="MathJax-Span-1050" style="font-family: STIXGeneral;"> </span><span class="mi" id="MathJax-Span-1051" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-1052" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.691em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.122em; border-left: 0px solid; width: 0px; height: 2.878em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo>∫</mo><mi>u</mi><mfrac><mrow><mi>d</mi><mi>v</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mtext> </mtext><mi>d</mi><mi>x</mi><mo>=</mo><mi>u</mi><mi>v</mi><mo>−</mo><mo>∫</mo><mi>v</mi><mfrac><mrow><mi>d</mi><mi>u</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mtext> </mtext><mi>d</mi><mi>x</mi></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-29">\int u \frac{dv}{dx} \ dx = uv - \int v \frac{du}{dx} \ dx</script></span></p><p>In general it is easier to make <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-30-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1053" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.51em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1054"><span class="mi" id="MathJax-Span-1055" style="font-family: STIXGeneral; font-style: italic;">u</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math></span></span><script type="math/tex" id="MathJax-Element-30">u</script></span> the simpler part of the given integrand.</p><h3 id="volume-of-a-revolution"><a class="anchor hidden-xs" href="#volume-of-a-revolution" title="volume-of-a-revolution"><span class="octicon octicon-link"></span></a>Volume of a revolution</h3><p>The volume of a function <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-31-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1056" style="width: 3.998em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.281em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1003.23em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1057"><span class="mi" id="MathJax-Span-1058" style="font-family: STIXGeneral; font-style: italic;">y</span><span class="mo" id="MathJax-Span-1059" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-1060" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-1061" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1062" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1063" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-31">y = f(x)</script></span> rotated fully around the <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-32-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1064" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1065"><span class="mi" id="MathJax-Span-1066" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span></span><script type="math/tex" id="MathJax-Element-32">x</script></span> axis is given by</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-33-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>V</mi><mo>=</mo><msubsup><mo>&#x222B;</mo><mi>a</mi><mi>b</mi></msubsup><mi>&#x03C0;</mi><msup><mi>y</mi><mn>2</mn></msup><mtext>&#xA0;</mtext><mi>d</mi><mi>x</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1067" style="width: 7.994em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.56em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.822em 1006.56em 3.64em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1068"><span class="mi" id="MathJax-Span-1069" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mo" id="MathJax-Span-1070" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="msubsup" id="MathJax-Span-1071" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.642em; height: 0px;"><span style="position: absolute; clip: rect(2.462em 1001.03em 5.023em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-1072" style="font-family: STIXIntegralsD; vertical-align: -0.612em;">∫<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.464em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -5.12em; left: 1.232em;"><span class="mi" id="MathJax-Span-1073" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">b</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.41em 4.152em -999.997em); top: -3.071em; left: 0.566em;"><span class="mi" id="MathJax-Span-1074" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mi" id="MathJax-Span-1075" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">π<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="msubsup" id="MathJax-Span-1076"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1077" style="font-family: STIXGeneral; font-style: italic;">y</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.464em;"><span class="mn" id="MathJax-Span-1078" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mtext" id="MathJax-Span-1079" style="font-family: STIXGeneral;"> </span><span class="mi" id="MathJax-Span-1080" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-1081" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.184em; border-left: 0px solid; width: 0px; height: 3.191em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>V</mi><mo>=</mo><msubsup><mo>∫</mo><mi>a</mi><mi>b</mi></msubsup><mi>π</mi><msup><mi>y</mi><mn>2</mn></msup><mtext> </mtext><mi>d</mi><mi>x</mi></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-33">V = \int _a ^b \pi y^2 \ dx</script></span></p><p>To find the volume of a revolution around the <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-34-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1082" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1083"><span class="mi" id="MathJax-Span-1084" style="font-family: STIXGeneral; font-style: italic;">y</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math></span></span><script type="math/tex" id="MathJax-Element-34">y</script></span> axis, rearrange the function to the form <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-35-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1085" style="width: 4.101em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.332em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1003.28em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1086"><span class="mi" id="MathJax-Span-1087" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1088" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-1089" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">g</span><span class="mo" id="MathJax-Span-1090" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1091" style="font-family: STIXGeneral; font-style: italic;">y</span><span class="mo" id="MathJax-Span-1092" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-35">x = g(y)</script></span>.</p><h2 id="numerical-solutions-and-iterative-methods"><a class="anchor hidden-xs" href="#numerical-solutions-and-iterative-methods" title="numerical-solutions-and-iterative-methods"><span class="octicon octicon-link"></span></a>Numerical solutions and iterative methods</h2><h3 id="iteration"><a class="anchor hidden-xs" href="#iteration" title="iteration"><span class="octicon octicon-link"></span></a>Iteration</h3><p>Iteration requires a formula in the form <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-36-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1093" style="width: 6.355em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.177em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1005.13em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1094"><span class="msubsup" id="MathJax-Span-1095"><span style="display: inline-block; position: relative; width: 1.898em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1096" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="texatom" id="MathJax-Span-1097"><span class="mrow" id="MathJax-Span-1098"><span class="mi" id="MathJax-Span-1099" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1100" style="font-size: 70.7%; font-family: STIXGeneral;">+</span><span class="mn" id="MathJax-Span-1101" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1102" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-1103" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-1104" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-1105"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1106" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mi" id="MathJax-Span-1107" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1108" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-36">X_{n+1} = f(x_n)</script></span></p><p>An iterative formula converges if <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-37-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msup><mi>f</mi><mo>&#x2032;</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>&lt;</mo><mn>1</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1109" style="width: 5.33em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.357em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.642em 1004.25em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1110"><span class="texatom" id="MathJax-Span-1111"><span class="mrow" id="MathJax-Span-1112"><span class="mo" id="MathJax-Span-1113" style="font-family: STIXVariants;">|</span></span></span><span class="msup" id="MathJax-Span-1114"><span style="display: inline-block; position: relative; width: 0.822em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1115" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.515em;"><span class="mo" id="MathJax-Span-1116" style="font-size: 70.7%; font-family: STIXVariants;">′</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1117" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1118" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1119" style="font-family: STIXGeneral;">)</span><span class="texatom" id="MathJax-Span-1120"><span class="mrow" id="MathJax-Span-1121"><span class="mo" id="MathJax-Span-1122" style="font-family: STIXVariants;">|</span></span></span><span class="mo" id="MathJax-Span-1123" style="font-family: STIXGeneral; padding-left: 0.31em;"><</span><span class="mn" id="MathJax-Span-1124" style="font-family: STIXGeneral; padding-left: 0.31em;">1</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.316em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msup><mi>f</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo><</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-37">|f'(x)| < 1</script></span> | for certain values of x</p><p>If the iteration converges to a limit the limit can be found by setting <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-38-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>x</mi><mi>n</mi></msub><mo>=</mo><mi>L</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1125" style="width: 7.021em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.74em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1005.74em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1126"><span class="msubsup" id="MathJax-Span-1127"><span style="display: inline-block; position: relative; width: 1.693em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1128" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="texatom" id="MathJax-Span-1129"><span class="mrow" id="MathJax-Span-1130"><span class="mi" id="MathJax-Span-1131" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1132" style="font-size: 70.7%; font-family: STIXGeneral;">+</span><span class="mn" id="MathJax-Span-1133" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1134" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="msubsup" id="MathJax-Span-1135" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1136" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mi" id="MathJax-Span-1137" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1138" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-1139" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>x</mi><mi>n</mi></msub><mo>=</mo><mi>L</mi></math></span></span><script type="math/tex" id="MathJax-Element-38">x_{n+1} = x_n = L</script></span></p><h3 id="trapezium-rule"><a class="anchor hidden-xs" href="#trapezium-rule" title="trapezium-rule"><span class="octicon octicon-link"></span></a>Trapezium rule</h3><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-39-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mo>&#x222B;</mo><mi>a</mi><mi>b</mi></msubsup><mi>y</mi><mtext>&#xA0;</mtext><mi>d</mi><mi>x</mi><mo>&#x2248;</mo><mfrac><mrow><mi>b</mi><mo>&#x2212;</mo><mi>a</mi></mrow><mi>n</mi></mfrac><mo stretchy="false">[</mo><msub><mi>y</mi><mn>0</mn></msub><mo>+</mo><msub><mi>y</mi><mi>n</mi></msub><mo>+</mo><mn>2</mn><mo stretchy="false">(</mo><msub><mi>y</mi><mn>1</mn></msub><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>+</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>&#x2212;</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1140" style="width: 22.697em; display: inline-block;"><span style="display: inline-block; position: relative; width: 18.599em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.822em 1018.5em 3.64em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1141"><span class="msubsup" id="MathJax-Span-1142"><span style="display: inline-block; position: relative; width: 1.642em; height: 0px;"><span style="position: absolute; clip: rect(2.462em 1001.03em 5.023em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-1143" style="font-family: STIXIntegralsD; vertical-align: -0.612em;">∫<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.464em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -5.12em; left: 1.232em;"><span class="mi" id="MathJax-Span-1144" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">b</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.41em 4.152em -999.997em); top: -3.071em; left: 0.566em;"><span class="mi" id="MathJax-Span-1145" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mi" id="MathJax-Span-1146" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">y</span><span class="mtext" id="MathJax-Span-1147" style="font-family: STIXGeneral;"> </span><span class="mi" id="MathJax-Span-1148" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-1149" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1150" style="font-family: STIXGeneral; padding-left: 0.31em;">≈</span><span class="mfrac" id="MathJax-Span-1151" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 2.308em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1002.21em 4.306em -999.997em); top: -4.659em; left: 50%; margin-left: -1.124em;"><span class="mrow" id="MathJax-Span-1152"><span class="mi" id="MathJax-Span-1153" style="font-family: STIXGeneral; font-style: italic;">b</span><span class="mo" id="MathJax-Span-1154" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-1155" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">a</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-1156" style="font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1002.31em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.308em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-1157" style="font-family: STIXGeneral;">[</span><span class="msubsup" id="MathJax-Span-1158"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1159" style="font-family: STIXGeneral; font-style: italic;">y</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mn" id="MathJax-Span-1160" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1161" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="msubsup" id="MathJax-Span-1162" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1163" style="font-family: STIXGeneral; font-style: italic;">y</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mi" id="MathJax-Span-1164" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1165" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mn" id="MathJax-Span-1166" style="font-family: STIXGeneral; padding-left: 0.259em;">2</span><span class="mo" id="MathJax-Span-1167" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-1168"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1169" style="font-family: STIXGeneral; font-style: italic;">y</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mn" id="MathJax-Span-1170" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1171" style="font-family: STIXGeneral;">+</span><span class="mo" id="MathJax-Span-1172" style="font-family: STIXGeneral;">.</span><span class="mo" id="MathJax-Span-1173" style="font-family: STIXGeneral; padding-left: 0.207em;">.</span><span class="mo" id="MathJax-Span-1174" style="font-family: STIXGeneral; padding-left: 0.207em;">.</span><span class="mo" id="MathJax-Span-1175" style="font-family: STIXGeneral; padding-left: 0.207em;">+</span><span class="msubsup" id="MathJax-Span-1176"><span style="display: inline-block; position: relative; width: 1.693em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1177" style="font-family: STIXGeneral; font-style: italic;">y</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="texatom" id="MathJax-Span-1178"><span class="mrow" id="MathJax-Span-1179"><span class="mi" id="MathJax-Span-1180" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1181" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1182" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1183" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1184" style="font-family: STIXGeneral;">]</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.184em; border-left: 0px solid; width: 0px; height: 3.191em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mo>∫</mo><mi>a</mi><mi>b</mi></msubsup><mi>y</mi><mtext> </mtext><mi>d</mi><mi>x</mi><mo>≈</mo><mfrac><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow><mi>n</mi></mfrac><mo stretchy="false">[</mo><msub><mi>y</mi><mn>0</mn></msub><mo>+</mo><msub><mi>y</mi><mi>n</mi></msub><mo>+</mo><mn>2</mn><mo stretchy="false">(</mo><msub><mi>y</mi><mn>1</mn></msub><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>+</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-39">\int _a ^b y\ dx \approx \frac{b-a}{n} [y_0 + y_n + 2(y_1 + ... + y_{n-1})]</script></span></p><p>If <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-40-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1185" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1186"><span class="mi" id="MathJax-Span-1187" style="font-family: STIXGeneral; font-style: italic;">y</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math></span></span><script type="math/tex" id="MathJax-Element-40">y</script></span> tends upwards the rule will overestimate, and if <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-41-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1188" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1189"><span class="mi" id="MathJax-Span-1190" style="font-family: STIXGeneral; font-style: italic;">y</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math></span></span><script type="math/tex" id="MathJax-Element-41">y</script></span> tends downwards the rule will underestimate.</p><h3 id="mid-ordinate-rule"><a class="anchor hidden-xs" href="#mid-ordinate-rule" title="mid-ordinate-rule"><span class="octicon octicon-link"></span></a>Mid-Ordinate rule</h3><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-42-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mo>&#x222B;</mo><mi>a</mi><mi>b</mi></msubsup><mi>y</mi><mtext>&#xA0;</mtext><mi>d</mi><mi>x</mi><mo>&#x2248;</mo><mfrac><mrow><mi>b</mi><mo>&#x2212;</mo><mi>a</mi></mrow><mi>n</mi></mfrac><mo stretchy="false">(</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msub><mo>+</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></msub><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>+</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>&#x2212;</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></msub><mo>+</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>&#x2212;</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msub><mo stretchy="false">)</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1191" style="width: 23.568em; display: inline-block;"><span style="display: inline-block; position: relative; width: 19.316em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.822em 1019.26em 3.64em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1192"><span class="msubsup" id="MathJax-Span-1193"><span style="display: inline-block; position: relative; width: 1.642em; height: 0px;"><span style="position: absolute; clip: rect(2.462em 1001.03em 5.023em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-1194" style="font-family: STIXIntegralsD; vertical-align: -0.612em;">∫<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.464em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -5.12em; left: 1.232em;"><span class="mi" id="MathJax-Span-1195" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">b</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.41em 4.152em -999.997em); top: -3.071em; left: 0.566em;"><span class="mi" id="MathJax-Span-1196" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mi" id="MathJax-Span-1197" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">y</span><span class="mtext" id="MathJax-Span-1198" style="font-family: STIXGeneral;"> </span><span class="mi" id="MathJax-Span-1199" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-1200" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1201" style="font-family: STIXGeneral; padding-left: 0.31em;">≈</span><span class="mfrac" id="MathJax-Span-1202" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 2.308em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1002.21em 4.306em -999.997em); top: -4.659em; left: 50%; margin-left: -1.124em;"><span class="mrow" id="MathJax-Span-1203"><span class="mi" id="MathJax-Span-1204" style="font-family: STIXGeneral; font-style: italic;">b</span><span class="mo" id="MathJax-Span-1205" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-1206" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">a</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-1207" style="font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1002.31em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.308em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-1208" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-1209"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1210" style="font-family: STIXGeneral; font-style: italic;">y</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.686em; left: 0.464em;"><span class="texatom" id="MathJax-Span-1211"><span class="mrow" id="MathJax-Span-1212"><span class="mfrac" id="MathJax-Span-1213"><span style="display: inline-block; position: relative; width: 0.361em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.486em 1000.21em 4.152em -999.997em); top: -4.301em; left: 50%; margin-left: -0.1em;"><span class="mn" id="MathJax-Span-1214" style="font-size: 50%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.486em 1000.26em 4.152em -999.997em); top: -3.686em; left: 50%; margin-left: -0.1em;"><span class="mn" id="MathJax-Span-1215" style="font-size: 50%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.36em 1.232em -999.997em); top: -1.227em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.361em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1216" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="msubsup" id="MathJax-Span-1217" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1218" style="font-family: STIXGeneral; font-style: italic;">y</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.635em; left: 0.464em;"><span class="texatom" id="MathJax-Span-1219"><span class="mrow" id="MathJax-Span-1220"><span class="mfrac" id="MathJax-Span-1221"><span style="display: inline-block; position: relative; width: 0.361em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.486em 1000.21em 4.152em -999.997em); top: -4.352em; left: 50%; margin-left: -0.1em;"><span class="mn" id="MathJax-Span-1222" style="font-size: 50%; font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.486em 1000.26em 4.152em -999.997em); top: -3.686em; left: 50%; margin-left: -0.1em;"><span class="mn" id="MathJax-Span-1223" style="font-size: 50%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.36em 1.232em -999.997em); top: -1.227em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.361em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1224" style="font-family: STIXGeneral;">+</span><span class="mo" id="MathJax-Span-1225" style="font-family: STIXGeneral;">.</span><span class="mo" id="MathJax-Span-1226" style="font-family: STIXGeneral; padding-left: 0.207em;">.</span><span class="mo" id="MathJax-Span-1227" style="font-family: STIXGeneral; padding-left: 0.207em;">.</span><span class="mo" id="MathJax-Span-1228" style="font-family: STIXGeneral; padding-left: 0.207em;">+</span><span class="msubsup" id="MathJax-Span-1229"><span style="display: inline-block; position: relative; width: 1.949em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1230" style="font-family: STIXGeneral; font-style: italic;">y</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.635em; left: 0.464em;"><span class="texatom" id="MathJax-Span-1231"><span class="mrow" id="MathJax-Span-1232"><span class="mi" id="MathJax-Span-1233" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1234" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mfrac" id="MathJax-Span-1235"><span style="display: inline-block; position: relative; width: 0.361em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.486em 1000.21em 4.152em -999.997em); top: -4.352em; left: 50%; margin-left: -0.1em;"><span class="mn" id="MathJax-Span-1236" style="font-size: 50%; font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.486em 1000.26em 4.152em -999.997em); top: -3.686em; left: 50%; margin-left: -0.1em;"><span class="mn" id="MathJax-Span-1237" style="font-size: 50%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.36em 1.232em -999.997em); top: -1.227em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.361em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1238" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="msubsup" id="MathJax-Span-1239" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 1.949em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1240" style="font-family: STIXGeneral; font-style: italic;">y</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.686em; left: 0.464em;"><span class="texatom" id="MathJax-Span-1241"><span class="mrow" id="MathJax-Span-1242"><span class="mi" id="MathJax-Span-1243" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1244" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mfrac" id="MathJax-Span-1245"><span style="display: inline-block; position: relative; width: 0.361em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.486em 1000.21em 4.152em -999.997em); top: -4.301em; left: 50%; margin-left: -0.1em;"><span class="mn" id="MathJax-Span-1246" style="font-size: 50%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.486em 1000.26em 4.152em -999.997em); top: -3.686em; left: 50%; margin-left: -0.1em;"><span class="mn" id="MathJax-Span-1247" style="font-size: 50%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.36em 1.232em -999.997em); top: -1.227em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.361em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1248" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.184em; border-left: 0px solid; width: 0px; height: 3.191em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mo>∫</mo><mi>a</mi><mi>b</mi></msubsup><mi>y</mi><mtext> </mtext><mi>d</mi><mi>x</mi><mo>≈</mo><mfrac><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow><mi>n</mi></mfrac><mo stretchy="false">(</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msub><mo>+</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></msub><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>+</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></msub><mo>+</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msub><mo stretchy="false">)</mo></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-42">\int _a ^b y\ dx \approx \frac{b-a}{n} (y_{\frac{1}{2}} + y_{\frac{3}{2}} + ... + y_{n - \frac{3}{2}} + y_{n - \frac{1}{2}}) </script></span></p><p>Where <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-43-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1249" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1250"><span class="mi" id="MathJax-Span-1251" style="font-family: STIXGeneral; font-style: italic;">n</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-43">n</script></span> is even.</p><h3 id="simpson’s-rule"><a class="anchor hidden-xs" href="#simpson’s-rule" title="simpson’s-rule"><span class="octicon octicon-link"></span></a>Simpson’s rule</h3><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-44-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mo>&#x222B;</mo><mi>a</mi><mi>b</mi></msubsup><mi>y</mi><mtext>&#xA0;</mtext><mi>d</mi><mi>x</mi><mo>&#x2248;</mo><mfrac><mrow><mi>b</mi><mo>&#x2212;</mo><mi>a</mi></mrow><mrow><mn>3</mn><mi>n</mi></mrow></mfrac><mo stretchy="false">[</mo><msub><mi>y</mi><mn>0</mn></msub><mo>+</mo><msub><mi>y</mi><mi>n</mi></msub><mo>+</mo><mn>4</mn><mo stretchy="false">(</mo><msub><mi>y</mi><mn>1</mn></msub><mo>+</mo><msub><mi>y</mi><mn>3</mn></msub><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>+</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>&#x2212;</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn><mo stretchy="false">(</mo><msub><mi>y</mi><mn>2</mn></msub><mo>+</mo><msub><mi>y</mi><mn>4</mn></msub><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>+</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>&#x2212;</mo><mn>2</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1252" style="width: 37.144em; display: inline-block;"><span style="display: inline-block; position: relative; width: 30.433em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.822em 1030.33em 3.64em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1253"><span class="msubsup" id="MathJax-Span-1254"><span style="display: inline-block; position: relative; width: 1.642em; height: 0px;"><span style="position: absolute; clip: rect(2.462em 1001.03em 5.023em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-1255" style="font-family: STIXIntegralsD; vertical-align: -0.612em;">∫<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.464em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -5.12em; left: 1.232em;"><span class="mi" id="MathJax-Span-1256" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">b</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.41em 4.152em -999.997em); top: -3.071em; left: 0.566em;"><span class="mi" id="MathJax-Span-1257" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mi" id="MathJax-Span-1258" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">y</span><span class="mtext" id="MathJax-Span-1259" style="font-family: STIXGeneral;"> </span><span class="mi" id="MathJax-Span-1260" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-1261" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1262" style="font-family: STIXGeneral; padding-left: 0.31em;">≈</span><span class="mfrac" id="MathJax-Span-1263" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 2.308em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1002.21em 4.306em -999.997em); top: -4.659em; left: 50%; margin-left: -1.124em;"><span class="mrow" id="MathJax-Span-1264"><span class="mi" id="MathJax-Span-1265" style="font-family: STIXGeneral; font-style: italic;">b</span><span class="mo" id="MathJax-Span-1266" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-1267" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">a</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.51em;"><span class="mrow" id="MathJax-Span-1268"><span class="mn" id="MathJax-Span-1269" style="font-family: STIXGeneral;">3</span><span class="mi" id="MathJax-Span-1270" style="font-family: STIXGeneral; font-style: italic;">n</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1002.31em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.308em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-1271" style="font-family: STIXGeneral;">[</span><span class="msubsup" id="MathJax-Span-1272"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1273" style="font-family: STIXGeneral; font-style: italic;">y</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mn" id="MathJax-Span-1274" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1275" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="msubsup" id="MathJax-Span-1276" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1277" style="font-family: STIXGeneral; font-style: italic;">y</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mi" id="MathJax-Span-1278" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1279" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mn" id="MathJax-Span-1280" style="font-family: STIXGeneral; padding-left: 0.259em;">4</span><span class="mo" id="MathJax-Span-1281" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-1282"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1283" style="font-family: STIXGeneral; font-style: italic;">y</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mn" id="MathJax-Span-1284" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1285" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="msubsup" id="MathJax-Span-1286" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1287" style="font-family: STIXGeneral; font-style: italic;">y</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mn" id="MathJax-Span-1288" style="font-size: 70.7%; font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1289" style="font-family: STIXGeneral;">+</span><span class="mo" id="MathJax-Span-1290" style="font-family: STIXGeneral;">.</span><span class="mo" id="MathJax-Span-1291" style="font-family: STIXGeneral; padding-left: 0.207em;">.</span><span class="mo" id="MathJax-Span-1292" style="font-family: STIXGeneral; padding-left: 0.207em;">.</span><span class="mo" id="MathJax-Span-1293" style="font-family: STIXGeneral; padding-left: 0.207em;">+</span><span class="msubsup" id="MathJax-Span-1294"><span style="display: inline-block; position: relative; width: 1.693em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1295" style="font-family: STIXGeneral; font-style: italic;">y</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="texatom" id="MathJax-Span-1296"><span class="mrow" id="MathJax-Span-1297"><span class="mi" id="MathJax-Span-1298" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1299" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1300" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1301" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1302" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mn" id="MathJax-Span-1303" style="font-family: STIXGeneral; padding-left: 0.259em;">2</span><span class="mo" id="MathJax-Span-1304" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-1305"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1306" style="font-family: STIXGeneral; font-style: italic;">y</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mn" id="MathJax-Span-1307" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1308" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="msubsup" id="MathJax-Span-1309" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1310" style="font-family: STIXGeneral; font-style: italic;">y</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mn" id="MathJax-Span-1311" style="font-size: 70.7%; font-family: STIXGeneral;">4</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1312" style="font-family: STIXGeneral;">+</span><span class="mo" id="MathJax-Span-1313" style="font-family: STIXGeneral;">.</span><span class="mo" id="MathJax-Span-1314" style="font-family: STIXGeneral; padding-left: 0.207em;">.</span><span class="mo" id="MathJax-Span-1315" style="font-family: STIXGeneral; padding-left: 0.207em;">.</span><span class="mo" id="MathJax-Span-1316" style="font-family: STIXGeneral; padding-left: 0.207em;">+</span><span class="msubsup" id="MathJax-Span-1317"><span style="display: inline-block; position: relative; width: 1.693em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1318" style="font-family: STIXGeneral; font-style: italic;">y</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="texatom" id="MathJax-Span-1319"><span class="mrow" id="MathJax-Span-1320"><span class="mi" id="MathJax-Span-1321" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1322" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1323" style="font-size: 70.7%; font-family: STIXGeneral;">2</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1324" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1325" style="font-family: STIXGeneral;">]</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.184em; border-left: 0px solid; width: 0px; height: 3.191em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mo>∫</mo><mi>a</mi><mi>b</mi></msubsup><mi>y</mi><mtext> </mtext><mi>d</mi><mi>x</mi><mo>≈</mo><mfrac><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow><mrow><mn>3</mn><mi>n</mi></mrow></mfrac><mo stretchy="false">[</mo><msub><mi>y</mi><mn>0</mn></msub><mo>+</mo><msub><mi>y</mi><mi>n</mi></msub><mo>+</mo><mn>4</mn><mo stretchy="false">(</mo><msub><mi>y</mi><mn>1</mn></msub><mo>+</mo><msub><mi>y</mi><mn>3</mn></msub><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>+</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn><mo stretchy="false">(</mo><msub><mi>y</mi><mn>2</mn></msub><mo>+</mo><msub><mi>y</mi><mn>4</mn></msub><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>+</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-44">\int _a ^b y\ dx \approx \frac{b-a}{3n} [y_0 + y_n + 4(y_1 + y_3 + ... + y_{n-1}) + 2(y_2 + y_4 + ... + y_{n-2})]</script></span></p><h3 id="cobweb-diagram"><a class="anchor hidden-xs" href="#cobweb-diagram" title="cobweb-diagram"><span class="octicon octicon-link"></span></a>Cobweb diagram</h3><p>Plot the two functions</p><p>A cobweb diagram is plotted by drawing up from the first x value, before rotating around, alternating between each of the functions for each value of x.</p><p>This is more appropriate when the values converge around the root rather than towards it.</p><p><img src="https://www.mathscard.co.uk/files/figures/41.png" alt=""></p><h3 id="staircase-diagram"><a class="anchor hidden-xs" href="#staircase-diagram" title="staircase-diagram"><span class="octicon octicon-link"></span></a>Staircase diagram</h3><p>Plot the two functions</p><p>A staircase is plotted by drawing a line up from each x value to one of the functions, and then drawing across to the next intersect with the other function.</p><p><img src="https://www.mathscard.co.uk/files/figures/42.png" alt=""></p></div>
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<div class="toc"><ul class="nav"><li class=""><a href="#core-3">Core 3</a><ul class="nav"><li class=""><a href="#functions">Functions</a><ul class="nav"><li class=""><a href="#domain-and-range-of-a-function">Domain and range of a function</a></li><li class=""><a href="#inverse-of-a-function">Inverse of a function</a></li></ul></li><li class=""><a href="#transformations-and-the-modulus-function">Transformations and the modulus function</a><ul class="nav"><li class=""><a href="#translations">Translations</a></li></ul></li><li><a href="#inverse-trigonometric-functions">Inverse trigonometric functions</a></li><li class=""><a href="#trig-identities">Trig identities</a></li><li class=""><a href="#calculus">Calculus</a><ul class="nav"><li class=""><a href="#log-rules">Log rules</a></li><li><a href="#differentials-and-integrals">Differentials and integrals</a></li><li><a href="#the-chain-rule">The chain rule</a></li><li class=""><a href="#the-product-rule">The product rule</a></li><li><a href="#the-quotient-rule">The quotient rule</a></li><li><a href="#integration-by-inspection-and-substitution">Integration by inspection and substitution</a></li><li class=""><a href="#integration-by-parts-and-standard-integrals">Integration by parts and standard integrals</a></li><li class=""><a href="#volume-of-a-revolution">Volume of a revolution</a></li></ul></li><li class=""><a href="#numerical-solutions-and-iterative-methods">Numerical solutions and iterative methods</a><ul class="nav"><li class=""><a href="#iteration">Iteration</a></li><li class=""><a href="#trapezium-rule">Trapezium rule</a></li><li class=""><a href="#mid-ordinate-rule">Mid-Ordinate rule</a></li><li class=""><a href="#simpson’s-rule">Simpson’s rule</a></li><li class=""><a href="#cobweb-diagram">Cobweb diagram</a></li><li class=""><a href="#staircase-diagram">Staircase diagram</a></li></ul></li></ul></li></ul></div><div class="toc-menu"><a class="expand-toggle" href="#">Expand all</a><a class="back-to-top" href="#">Back to top</a><a class="go-to-bottom" href="#">Go to bottom</a></div>
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<div class="toc"><ul class="nav"><li class=""><a href="#core-3">Core 3</a><ul class="nav"><li class=""><a href="#functions">Functions</a><ul class="nav"><li class=""><a href="#domain-and-range-of-a-function">Domain and range of a function</a></li><li class=""><a href="#inverse-of-a-function">Inverse of a function</a></li></ul></li><li class=""><a href="#transformations-and-the-modulus-function">Transformations and the modulus function</a><ul class="nav"><li class=""><a href="#translations">Translations</a></li></ul></li><li><a href="#inverse-trigonometric-functions">Inverse trigonometric functions</a></li><li class=""><a href="#trig-identities">Trig identities</a></li><li class=""><a href="#calculus">Calculus</a><ul class="nav"><li class=""><a href="#log-rules">Log rules</a></li><li><a href="#differentials-and-integrals">Differentials and integrals</a></li><li><a href="#the-chain-rule">The chain rule</a></li><li class=""><a href="#the-product-rule">The product rule</a></li><li><a href="#the-quotient-rule">The quotient rule</a></li><li><a href="#integration-by-inspection-and-substitution">Integration by inspection and substitution</a></li><li class=""><a href="#integration-by-parts-and-standard-integrals">Integration by parts and standard integrals</a></li><li class=""><a href="#volume-of-a-revolution">Volume of a revolution</a></li></ul></li><li class=""><a href="#numerical-solutions-and-iterative-methods">Numerical solutions and iterative methods</a><ul class="nav"><li class=""><a href="#iteration">Iteration</a></li><li class=""><a href="#trapezium-rule">Trapezium rule</a></li><li class=""><a href="#mid-ordinate-rule">Mid-Ordinate rule</a></li><li class=""><a href="#simpson’s-rule">Simpson’s rule</a></li><li class=""><a href="#cobweb-diagram">Cobweb diagram</a></li><li class=""><a href="#staircase-diagram">Staircase diagram</a></li></ul></li></ul></li></ul></div><div class="toc-menu"><a class="expand-toggle" href="#">Expand all</a><a class="back-to-top" href="#">Back to top</a><a class="go-to-bottom" href="#">Go to bottom</a></div>
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