To integrate $\sin(4x) \cos^2(4x)$, we can simplify the expression by using trigonometric identities. Here's the step-by-step solution:

1. Rewrite $\cos^2(4x)$ using the identity: $\cos^2(4x) = \frac{1 + \cos(8x)}{2}$ So the integral becomes: $\int \sin(4x) \cos^2(4x) \, dx = \int \sin(4x) \cdot \frac{1 + \cos(8x)}{2} \, dx$

2. Distribute $\sin(4x)$ across the terms: $= \frac{1}{2} \int \sin(4x) \, dx + \frac{1}{2} \int \sin(4x) \cos(8x) \, dx$

3. Integrate each term separately.

First Integral

For $\frac{1}{2} \int \sin(4x) \, dx$:

• The antiderivative of $\sin(4x)$ is $-\frac{1}{4} \cos(4x)$, so: $\frac{1}{2} \int \sin(4x) \, dx = \frac{1}{2} \cdot \left( -\frac{1}{4} \cos(4x) \right) = -\frac{1}{8} \cos(4x)$

Second Integral

For $\frac{1}{2} \int \sin(4x) \cos(8x) \, dx$, we use the product-to-sum identity: $\sin(4x) \cos(8x) = \frac{1}{2} \left( \sin(12x) + \sin(-4x) \right)$ Substituting, we get: $\frac{1}{2} \int \sin(4x) \cos(8x) \, dx = \frac{1}{4} \int \left( \sin(12x) - \sin(4x) \right) dx$

Now we integrate each part separately:

• The integral of $\sin(12x)$ is $-\frac{1}{12} \cos(12x)$.
• The integral of $-\sin(4x)$ is $\frac{1}{4} \cos(4x)$.

Putting these together: $\frac{1}{4} \int \left( \sin(12x) - \sin(4x) \right) dx = \frac{1}{4} \left( -\frac{1}{12} \cos(12x) + \frac{1}{4} \cos(4x) \right)$ Simplifying, we get: $= -\frac{1}{48} \cos(12x) + \frac{1}{16} \cos(4x)$

Final Answer

Combining the results, we get: $\int \sin(4x) \cos^2(4x) \, dx = -\frac{1}{8} \cos(4x) - \frac{1}{48} \cos(12x) + \frac{1}{16} \cos(4x) + C$ Simplify the terms involving $\cos(4x)$: $= -\frac{1}{48} \cos(12x) - \frac{1}{16} \cos(4x) + C$

Let me know if you’d like further clarification or details on any of these steps!

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Additional Questions:

1. How would you approach integrating functions involving higher powers of trigonometric functions?
2. What are the product-to-sum identities, and how can they be applied to integration problems?
3. How does simplifying trigonometric integrals benefit from using identities?
4. What would the integration look like if we had $\sin^2(4x) \cos(4x)$ instead?
5. What techniques are useful for integrating products of sine and cosine functions with different arguments?

Tip:

Applying trigonometric identities can often simplify complex integrals, making them easier to solve directly.