Example 8

∫ sin⁴ x cos³x dx.

Let u = sin x, du = cos x dx.

∫ sin⁴x cos³x dx = ∫ u⁴(l - u²)du

= (1/5) u⁵ - (1/7) u⁷ + c

= (1/5) sin⁵ x - (1/7) sin⁷ x + C.

This method also works for an odd power of sin x times any power of cos x, and vice versa.