https://file.aiquickdraw.com/mathbot/file/1728219042805zaybgt4r.jpg

To solve the integral:

\[
\int \frac{\sin t}{(5 + \cos t)^3} \, dt
\]

we can use substitution to simplify it. Let's denote:

\[
u = 5 + \cos t
\]

### Step-by-Step Solution

1. **Substitute \( u = 5 + \cos t \)**

   Now we need to find the differential \( du \). Since \( u = 5 + \cos t \), differentiating both sides with respect to \( t \):

   \[
   \frac{du}{dt} = -\sin t
   \]

   Therefore:

   \[
   du = -\sin t \, dt
   \]

   Which implies:

   \[
   \sin t \, dt = -du
   \]

2. **Substitute into the Integral**

   Now replace in the integral:

   \[
   \int \frac{\sin t}{(5 + \cos t)^3} \, dt = \int \frac{-du}{u^3}
   \]

3. **Simplify the Integral**

   The integral now becomes:

   \[
   \int -u^{-3} \, du = -\int u^{-3} \, du
   \]

4. **Integrate**

   To integrate \( u^{-3} \):

   \[
   -\int u^{-3} \, du = -\frac{u^{-2}}{-2} = \frac{1}{2} u^{-2} = \frac{1}{2} \cdot \frac{1}{u^2}
   \]

5. **Substitute Back \( u = 5 + \cos t \)**

   Now replace \( u \) back with \( 5 + \cos t \):

   \[
   \frac{1}{2} \cdot \frac{1}{(5 + \cos t)^2} = \frac{1}{2(5 + \cos t)^2}
   \]

6. **Add the Constant of Integration**

   Don't forget to add the constant of integration, \( C \):

   \[
   \frac{1}{2(5 + \cos t)^2} + C
   \]

### Matching with Given Options

The correct answer matches option **B**:

\[
\boxed{\frac{1}{2(5 + \cos t)^2} + C}
\]

---

Would you like more details or have any questions on this solution? 

Here are some related questions you might find interesting:

1. How do you decide which substitution to use in an integral problem?
2. What are the conditions under which substitution is an effective method for integration?
3. Can you explain the power rule for integration in more detail?
4. How do trigonometric substitutions differ from standard algebraic substitutions?
5. What are other methods of integration besides substitution?

**Tip:** Always look out for a composition of functions in an integral—it's a good indicator that substitution might simplify the problem.

Evaluate the integral: \( \int \frac{\sin t}{(5 + \cos t)^3} dt \)

Learn how to evaluate the integral \( \int \frac{\sin t}{(5 + \cos t)^3} dt \) using substitution. This problem involves key calculus concepts such as trigonometric substitution and the power rule for integration. Suitable for undergraduate calculus students.

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