To integrate $\cos^2(4x)$, we can use the power-reduction identity:

$\cos^2(4x) = \frac{1 + \cos(8x)}{2}$

This simplifies the integral:

$\int \cos^2(4x) \, dx = \int \frac{1 + \cos(8x)}{2} \, dx$

Now, we can separate the integral:

$= \frac{1}{2} \int 1 \, dx + \frac{1}{2} \int \cos(8x) \, dx$

Integrating each term separately:

1. For the first term: $\frac{1}{2} \int 1 \, dx = \frac{x}{2}$

2. For the second term: $\frac{1}{2} \int \cos(8x) \, dx = \frac{1}{2} \cdot \frac{\sin(8x)}{8} = \frac{\sin(8x)}{16}$

Combining these results:

$\int \cos^2(4x) \, dx = \frac{x}{2} + \frac{\sin(8x)}{16} + C$

where $C$ is the constant of integration.

Would you like further details on any steps?

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1. How is the power-reduction identity for cosine derived?
2. What is the significance of using substitution in integrals like this?
3. How would this integral change if we had $\cos^3(4x)$ instead?
4. How does the power-reduction identity help in integration of trigonometric powers?
5. What are the practical applications of integrals involving trigonometric functions?

Tip: Using trigonometric identities can greatly simplify integration of trigonometric expressions.