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.

Learn how to integrate cos^2(4x) with a step-by-step solution using the power-reduction identity. This problem demonstrates the application of trigonometric identities in integration, breaking down each step for Grades 11-12 students studying advanced trigonometry and calculus.

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