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!