How to integrate cos^2(x)sin(x)?

To integrate cos^2(x)sin(x), use the substitution u = cos(x) and the formula ∫u^2 du = u^3/3 + C.

To integrate cos^2(x)sin(x), use the substitution u = cos(x). Then, du/dx = -sin(x) and dx = -du/sin(x). Substituting these into the integral gives:

∫cos^2(x)sin(x) dx = ∫cos^2(x)sin(x) (-du/sin(x))
= -∫cos^2(x) du

Using the identity cos^2(x) = 1 - sin^2(x), we can rewrite the integral as:

-∫(1 - sin^2(x)) du = -∫du + ∫sin^2(x) du
= -u + ∫sin^2(x) du + C
= -cos(x) + ∫(1 - cos^2(x)) du + C
= -cos(x) + u - u^3/3 + C
= -cos(x) + cos^3(x)/3 + C

Therefore, the integral of cos^2(x)sin(x) is -cos(x) + cos^3(x)/3 + C.