How to integrate cos^3(x)?

To integrate cos^3(x), use the substitution u = sin(x) and the identity cos^2(x) = 1 - sin^2(x).

First, rewrite cos^3(x) as cos^2(x) * cos(x). Then, use the identity cos^2(x) = 1 - sin^2(x) to get:

cos^3(x) = cos^2(x) * cos(x) = (1 - sin^2(x)) * cos(x)

Next, substitute u = sin(x) and du/dx = cos(x) dx to get:

cos^3(x) dx = (1 - u^2) du

Now, integrate both sides with respect to u:

∫ cos^3(x) dx = ∫ (1 - u^2) du

= u - (1/3)u^3 + C

Substitute back in u = sin(x) to get the final answer:

∫ cos^3(x) dx = sin(x) - (1/3)sin^3(x) + C