https://file.aiquickdraw.com/mathbot/file/17284838448072o0bck4o.jpg

The integral provided is:

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

### Solution:

We can solve this integral using substitution. Let’s use the following substitution:

Let:
\[
u = 5 + \cos t
\]

Then, differentiate \( u \) with respect to \( t \):
\[
\frac{du}{dt} = -\sin t
\]
\[
dt = \frac{du}{-\sin t}
\]

Now substitute \( u = 5 + \cos t \) and \( dt = \frac{du}{-\sin t} \) into the integral:

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

The \( \sin t \) terms cancel out, and we are left with:

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

This is a simple power integral, which we can solve as follows:

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

Now, substitute back \( u = 5 + \cos t \):

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

### Final Answer:

\[
\int \frac{\sin t}{(5 + \cos t)^3} \, dt = \frac{1}{2(5 + \cos t)^2} + C
\]

Let me know if you need further details or clarifications!

---

Here are 5 related questions to deepen your understanding:
1. How does substitution simplify integrals involving trigonometric functions?
2. What happens if we substitute a different variable for \( u \)?
3. How can you integrate more complex functions involving sine and cosine?
4. What are the general rules for integrating functions with powers of trigonometric expressions?
5. How does the constant of integration affect the final solution?

**Tip:** When solving integrals, always check for symmetry or substitution options to simplify the problem!

Explore the detailed solution to the integral ∫ (sin t) / (5 + cos t)^3 dt using trigonometric substitution. This calculus problem is solved using u-substitution and the power rule, suitable for undergraduate students. Step-by-step explanation provided.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation