Evaluate the integral of cos^4(x) dx.

The integral of cos^4(x) dx is (3/8)x + (1/4)sin(2x) + (1/32)sin(4x) + C.

To evaluate the integral of cos^4(x) dx, we can use the identity cos^2(x) = (1/2)(1 + cos(2x)) to rewrite the integrand as cos^4(x) = (1/4)(1 + cos(2x))^2. Then, we can expand the square using the binomial theorem to get cos^4(x) = (1/4)(1 + 2cos(2x) + cos^2(2x)).

Next, we can use the identity sin^2(x) = 1 - cos^2(x) to rewrite cos^2(2x) as 1 - sin^2(2x). Substituting this into our expression for cos^4(x), we get cos^4(x) = (1/4)(1 + 2cos(2x) + 1 - sin^2(2x)) = (1/2)(1 + cos(2x)) + (1/8)(1 - sin^2(2x)).

Now, we can integrate term by term to get the final answer:

∫cos^4(x) dx = (1/2)∫(1 + cos(2x)) dx + (1/8)∫(1 - sin^2(2x)) dx

= (1/2)x + (1/4)sin(2x) + (1/32)sin(4x) + C,

where C is the constant of integration.