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.