To find the integral of \( \cos^4(x) \), we can use a power-reduction identity and some trigonometric identities to simplify the expression.

### Step 1: Use the power-reduction formula for \( \cos^2(x) \)
We start by rewriting \( \cos^4(x) \) as:
\[
\cos^4(x) = \left( \cos^2(x) \right)^2
\]
Using the identity for \( \cos^2(x) \):
\[
\cos^2(x) = \frac{1 + \cos(2x)}{2}
\]
We substitute this into the equation:
\[
\cos^4(x) = \left( \frac{1 + \cos(2x)}{2} \right)^2
\]

### Step 2: Expand the square
Now, expand the square:
\[
\cos^4(x) = \frac{1}{4} \left( 1 + 2\cos(2x) + \cos^2(2x) \right)
\]

### Step 3: Apply the power-reduction formula to \( \cos^2(2x) \)
We can apply the power-reduction identity again to \( \cos^2(2x) \):
\[
\cos^2(2x) = \frac{1 + \cos(4x)}{2}
\]
Substitute this back:
\[
\cos^4(x) = \frac{1}{4} \left( 1 + 2\cos(2x) + \frac{1 + \cos(4x)}{2} \right)
\]

### Step 4: Simplify the expression
Simplify the expression:
\[
\cos^4(x) = \frac{1}{4} \left( 1 + 2\cos(2x) + \frac{1}{2} + \frac{\cos(4x)}{2} \right)
\]
\[
\cos^4(x) = \frac{1}{4} \left( \frac{3}{2} + 2\cos(2x) + \frac{\cos(4x)}{2} \right)
\]
\[
\cos^4(x) = \frac{3}{8} + \frac{1}{2} \cos(2x) + \frac{1}{8} \cos(4x)
\]

### Step 5: Integrate term by term
Now, we can integrate each term:
\[
\int \cos^4(x) \, dx = \int \left( \frac{3}{8} + \frac{1}{2} \cos(2x) + \frac{1}{8} \cos(4x) \right) dx
\]
\[
= \frac{3}{8}x + \frac{1}{4} \sin(2x) + \frac{1}{32} \sin(4x) + C
\]

### Final Answer:
\[
\int \cos^4(x) \, dx = \frac{3}{8}x + \frac{1}{4} \sin(2x) + \frac{1}{32} \sin(4x) + C
\]

Would you like more details on any specific step, or have any questions?

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Here are 5 related questions to further your understanding:
1. How would you integrate \( \cos^6(x) \)?
2. What is the general process to integrate \( \cos^n(x) \) for any \( n \)?
3. How would you apply a similar approach to integrate \( \sin^4(x) \)?
4. Can you explain how trigonometric power-reduction identities are derived?
5. How would you solve a definite integral involving \( \cos^4(x) \), such as \( \int_0^{\pi} \cos^4(x) \, dx \)?

**Tip:** Power-reduction identities simplify the integration of even powers of sine and cosine functions, reducing higher powers into sums of lower powers with manageable integrals.

Explore the step-by-step solution to integrate cos^4(x) using power-reduction and trigonometric identities. This detailed guide walks through the simplification process, including key formulas and integrals, ideal for college-level calculus students.

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