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!