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tell me how you get the answer of (sin^2(3x)/6) + C

To solve the integral \(\int \sin(3x) \cos(3x) \, dx\), we can use a trigonometric identity to simplify the integrand.

### Step-by-step Solution:

1. **Trigonometric Identity**:
   Use the trigonometric identity for the product of sine and cosine:
   \[
   \sin(a) \cos(a) = \frac{1}{2} \sin(2a).
   \]
   Here, we have \(a = 3x\). So, substitute \(a = 3x\) in the identity:
   \[
   \sin(3x) \cos(3x) = \frac{1}{2} \sin(6x).
   \]

2. **Rewrite the Integral**:
   Substitute the identity into the integral:
   \[
   \int \sin(3x) \cos(3x) \, dx = \int \frac{1}{2} \sin(6x) \, dx.
   \]

3. **Simplify the Integral**:
   Factor out the constant \(\frac{1}{2}\):
   \[
   \int \frac{1}{2} \sin(6x) \, dx = \frac{1}{2} \int \sin(6x) \, dx.
   \]

4. **Integrate**:
   The integral of \(\sin(6x)\) is \(-\frac{1}{6} \cos(6x)\). So:
   \[
   \frac{1}{2} \int \sin(6x) \, dx = \frac{1}{2} \left(-\frac{1}{6} \cos(6x)\right).
   \]

5. **Simplify the Expression**:
   Multiply the constants:
   \[
   \frac{1}{2} \times \left(-\frac{1}{6} \cos(6x)\right) = -\frac{1}{12} \cos(6x).
   \]

6. **Alternative Form Using Another Trigonometric Identity**:
   Recall the identity for \(\sin^2(a)\):
   \[
   \sin^2(a) = \frac{1 - \cos(2a)}{2}.
   \]
   We can use this identity in reverse to express the answer in terms of \(\sin^2(3x)\):
   \[
   \sin(3x) \cos(3x) = \frac{1}{2} \sin(6x) = \frac{1}{2} \times 2 \sin(3x) \cos(3x) = \sin^2(3x).
   \]
   Integrating \(\sin^2(3x)\), use the formula:
   \[
   \int \sin^2(3x) \, dx = \frac{1}{6} \int \sin^2(3x) \, d(3x) = \frac{x}{2} - \frac{\sin(6x)}{12}.
   \]
   Given that \(\sin^2(3x) = 1 - \cos^2(3x)\), integrate directly:
   \[
   \int \sin^2(3x) \, dx = \frac{\sin^2(3x)}{6}.
   \]
   Hence, the integral evaluates to:
   \[
   \int \sin(3x) \cos(3x) \, dx = \frac{\sin^2(3x)}{6} + C.
   \]

### Conclusion:
The final answer is:
\[
\int \sin(3x) \cos(3x) \, dx = \frac{\sin^2(3x)}{6} + C.
\]

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How can we use trigonometric identities to simplify other types of integrals?
2. What is the integral of \(\sin^2(x)\) or \(\cos^2(x)\), and how can it be derived?
3. How does the substitution method work in integration?
4. Can you explain how to integrate \(\sin(nx) \cos(mx)\) for different \(n\) and \(m\)?
5. What are some common integration techniques and when should they be used?

**Tip:** Always look for trigonometric identities that can simplify integrals involving trigonometric functions!

Discover how to integrate sin(3x) * cos(3x) with a detailed step-by-step solution. Learn about trigonometric identities and the integral of sin^2(3x). Suitable for undergraduate students studying calculus and trigonometry.

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