To integrate \( \sin^5(3x) \cos(3x) \), we can use a substitution method and the fact that the sine and cosine functions are related. Here's how:

1. **Substitute**:
   Let \( u = \sin(3x) \), so that \( du = 3\cos(3x)dx \), or \( \frac{du}{3} = \cos(3x)dx \).

2. **Rewrite the integral**:
   The integral becomes:
   \[
   \int \sin^5(3x) \cos(3x) dx = \int u^5 \cdot \frac{du}{3}
   \]
   Simplify the constants:
   \[
   \frac{1}{3} \int u^5 du
   \]

3. **Integrate**:
   Now, integrate \( u^5 \):
   \[
   \frac{1}{3} \cdot \frac{u^6}{6} = \frac{u^6}{18}
   \]

4. **Substitute back** \( u = \sin(3x) \):
   \[
   \frac{(\sin(3x))^6}{18} + C
   \]

Thus, the integral is:
\[
\int \sin^5(3x) \cos(3x) dx = \frac{\sin^6(3x)}{18} + C
\]

### Do you want more details or have any questions?

Here are some related questions:
1. How would you integrate \( \sin^3(3x) \cos(3x) \)?
2. What happens when you integrate \( \cos^5(3x) \sin(3x) \)?
3. Can you solve the integral \( \sin^7(3x) \cos(3x) dx \)?
4. How would you apply substitution to solve \( \sin(3x) \cos^5(3x) dx \)?
5. What is the integral of \( \cos(3x) \sin^3(3x) dx \)?

**Tip**: When you see products of sine and cosine powers, substitution and using trigonometric identities often simplifies the process.

This page provides a step-by-step guide to solving the integral of sin^5(3x)cos(3x) using substitution. Learn how to apply the substitution method, rewrite the integral, and perform the integration for this trigonometric function. Suitable for students in Grades 11-12 or early college.

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