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?

---

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!