Update Taskiq with FastAPI documentation to include lifespan example

Author: Shaxzodbek16

docs/framework_integrations/taskiq-with-fastapi.md

@@ -19,7 +19,6 @@ import taskiq_fastapi
 broker = ZeroMQBroker()
 
 taskiq_fastapi.init(broker, "my_package.application:app")
-
 ```
 
 There are two rules to make everything work as you expect:
@@ -43,7 +42,6 @@ from typing import Any
 
 def get_redis_pool(request: Request) -> Any:
     return request.app.state.redis_pool
-
 ```
 
 To make it resolvable in taskiq, people should mark default fastapi dependencies (such as `Request` and `HTTPConnection`) with `TaskiqDepends`. Like this:
@@ -61,7 +59,6 @@ from taskiq import TaskiqDepends
 
 async def get_redis_pool(request: Annotated[Request, TaskiqDepends()]):
     return request.app.state.redis_pool
-
 ```
 
 @tab default values
@@ -73,14 +70,39 @@ from taskiq import TaskiqDepends
 
 async def get_redis_pool(request: Request = TaskiqDepends()):
     return request.app.state.redis_pool
-
 ```
 
 :::
 
 
 Also you want to call startup of your brokers somewhere.
 
+::: tabs
+
+@tab Lifespan (Recommended)
+
+```python
+from contextlib import asynccontextmanager
+from fastapi import FastAPI
+from your_project.taskiq import broker
+
+
+@asynccontextmanager
+async def lifespan(app: FastAPI):
+    # Startup
+    if not broker.is_worker_process:
+        await broker.startup()
+    yield
+    # Shutdown
+    if not broker.is_worker_process:
+        await broker.shutdown()
+
+
+app = FastAPI(lifespan=lifespan)
+```
+
+@tab on_event (Deprecated)
+
 ```python
 from fastapi import FastAPI
 from your_project.taskiq import broker
@@ -89,7 +111,7 @@ app = FastAPI()
 
 
 @app.on_event("startup")
-async def app_startup():
+asynchronous def app_startup():
     if not broker.is_worker_process:
         await broker.startup()
 
@@ -98,9 +120,10 @@ async def app_startup():
 async def app_shutdown():
     if not broker.is_worker_process:
         await broker.shutdown()
-
 ```
 
+:::
+
 And that's it. Now you can use your taskiq tasks with functions and classes that depend on FastAPI dependencies. You can find bigger examples in the [examples repo](https://github.com/taskiq-python/examples/).
 
 
@@ -114,7 +137,6 @@ Let's imagine that you have a fixture of your application. It returns a new fast
 @pytest.fixture
 def fastapi_app() -> FastAPI:
     return get_app()
-
 ```
 
 Right after this fixture, we define another one.
@@ -133,7 +155,6 @@ def init_taskiq_deps(fastapi_app: FastAPI):
     yield
 
     broker.custom_dependency_context = {}
-
 ```
 
 This fixture has autouse flag, which means it would run on every test automatically.