https://file.aiquickdraw.com/mathbot/file/1726509692615dqa65b27.jpg

The image shows an integral expression that can be interpreted as:

\[
\int \frac{dx}{\sin{x} \cdot \sin{2x}}
\]

To solve this integral, let's rewrite the expression for simplification.

### Step-by-Step Solution:

1. **Use Trigonometric Identities**:
   \[
   \sin{2x} = 2\sin{x}\cos{x}
   \]
   This will simplify the integral as:
   \[
   \int \frac{dx}{\sin{x} \cdot 2 \sin{x} \cos{x}} = \int \frac{dx}{2 \sin^2{x} \cos{x}}
   \]

2. **Substitute \(\sin{x}\)**:
   Let \( u = \sin{x} \), then \( du = \cos{x}dx \).

   The integral becomes:
   \[
   \int \frac{du}{2u^2}
   \]

3. **Integrate**:
   The integral of \(\frac{1}{u^2}\) is \(-\frac{1}{u}\), so:
   \[
   \int \frac{du}{2u^2} = -\frac{1}{2u} = -\frac{1}{2 \sin{x}}
   \]

Thus, the solution to the integral is:
\[
-\frac{1}{2 \sin{x}} + C
\]
where \(C\) is the constant of integration.

Let me know if you'd like more details or if you have any questions! Here are five related questions you might consider:

1. How do you solve integrals using substitution?
2. Can you apply other trigonometric identities to simplify different integrals?
3. What is the significance of the constant of integration in indefinite integrals?
4. How do definite integrals differ from indefinite ones?
5. What are the common trigonometric substitutions used in integrals?

**Tip:** Always look for trigonometric identities when dealing with integrals involving trigonometric functions. They often simplify the integral significantly!

Solve the integral \( \int \frac{dx}{\sin{x} \cdot \sin{2x}} \).

Learn how to solve the integral \( \int \frac{dx}{\sin{x} \cdot \sin{2x}} \) step-by-step using trigonometric substitution. The solution covers important concepts like trigonometric identities and basic integration rules. This problem is suitable for undergraduate calculus students.

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