https://file.aiquickdraw.com/mathbot/file/1726473193518hookywz5.jpg

The given expression in the image is an integral:

\[
\int \frac{dx}{\sin(x)(1 + 2 \cos(x))}
\]

This is an integral of a rational function involving sine and cosine. A common approach to solve integrals like this is to use substitution or trigonometric identities to simplify the expression.

### Let's try to solve it:

1. **Substitute \( u = \cos(x) \):**
   - Then, \( du = -\sin(x) dx \), so the integral becomes:
   
   \[
   \int \frac{-du}{(1 + 2u)}
   \]

2. **Simplify the integral:**

   \[
   -\int \frac{du}{1 + 2u}
   \]

3. **Solve the integral:**
   This is a basic logarithmic integral. The solution is:

   \[
   -\frac{1}{2} \ln |1 + 2u| + C
   \]

4. **Substitute back \( u = \cos(x) \):**

   \[
   -\frac{1}{2} \ln |1 + 2\cos(x)| + C
   \]

Thus, the solution to the integral is:

\[
-\frac{1}{2} \ln |1 + 2 \cos(x)| + C
\]

---

Would you like a more detailed explanation of any step or further clarification? 

Here are 5 related questions for further practice:

1. How do you solve integrals involving trigonometric functions like \(\sin(x)\) and \(\cos(x)\)?
2. Can you solve the integral \(\int \frac{dx}{1 + \cos(x)}\)?
3. What are common substitutions for integrals involving trigonometric functions?
4. How would you approach the integral \(\int \frac{dx}{\sin(x) (1 - \cos(x))}\)?
5. What is the method to integrate rational functions of sine and cosine?

**Tip:** Substituting trigonometric functions can often simplify complex-looking integrals!

This page provides a step-by-step solution for solving the integral ∫ dx / [sin(x)(1 + 2 cos(x))] using trigonometric substitution and logarithmic integration. Learn how substitution and integral properties simplify the solution.

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