To integrate \( \cos^4(x) \, dx \), we can use trigonometric identities to simplify the expression. Here's how we approach it step by step:

### Step 1: Use the power-reduction identity
We can express \( \cos^4(x) \) in terms of lower powers of cosine using the power-reduction identity:
\[
\cos^2(x) = \frac{1 + \cos(2x)}{2}
\]
So,
\[
\cos^4(x) = (\cos^2(x))^2 = \left( \frac{1 + \cos(2x)}{2} \right)^2
\]

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

### Step 3: Substitute and simplify the integral
Now we can split the integral into separate parts:
\[
\int \cos^4(x) \, dx = \frac{1}{4} \int \left( 1 + 2\cos(2x) + \cos^2(2x) \right) \, dx
\]

### Step 4: Integrate each term separately
1. **First term:**
   \[
   \frac{1}{4} \int 1 \, dx = \frac{x}{4}
   \]

2. **Second term:**
   \[
   \frac{1}{4} \int 2 \cos(2x) \, dx = \frac{1}{2} \int \cos(2x) \, dx = \frac{1}{2} \cdot \frac{\sin(2x)}{2} = \frac{\sin(2x)}{4}
   \]

3. **Third term:**
   To integrate \( \cos^2(2x) \), use the power-reduction identity again:
   \[
   \cos^2(2x) = \frac{1 + \cos(4x)}{2}
   \]
   So the integral becomes:
   \[
   \frac{1}{4} \int \cos^2(2x) \, dx = \frac{1}{4} \int \frac{1 + \cos(4x)}{2} \, dx = \frac{1}{8} \int (1 + \cos(4x)) \, dx
   \]
   Now integrate both parts:
   \[
   \frac{1}{8} \left( x + \frac{\sin(4x)}{4} \right) = \frac{x}{8} + \frac{\sin(4x)}{32}
   \]

### Step 5: Combine all parts
Now, combining all the integrated terms:
\[
\int \cos^4(x) \, dx = \frac{x}{4} + \frac{\sin(2x)}{4} + \frac{x}{8} + \frac{\sin(4x)}{32} + C
\]
Simplifying:
\[
\int \cos^4(x) \, dx = \frac{3x}{8} + \frac{\sin(2x)}{4} + \frac{\sin(4x)}{32} + C
\]

This is the final answer.

Would you like further clarification on any step?

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Here are some related questions for further exploration:
1. How would you integrate \( \cos^6(x) \, dx \)?
2. What are the power-reduction identities for higher powers of sine or cosine?
3. How do you solve integrals involving trigonometric functions raised to odd powers?
4. How can you derive the power-reduction formulas?
5. Can this method be used to integrate functions like \( \sin^4(x) \, dx \)?

**Tip:** Always consider using trigonometric identities for simplifying integrals involving powers of sine and cosine!

Learn how to integrate cos^4(x) step-by-step by applying the power-reduction identity. This guide simplifies the integration process with clear examples and detailed explanations, making it ideal for high school and college calculus students.

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