\int\limits^0_\pi {x*sin^{m} (x)} \, dx

Let

Integrate by parts, taking

u = x   ==>   du = dx

dv = sinᵐ (x) dx   ==>   v = ∫ sinᵐ (x) dx

so that

There is a well-known power reduction formula for this integral. If you want to derive it for yourself, consider the cases where m is even or where m is odd.

If m is even, then m = 2k for some integer k, and we have

Expand the binomial, then use the half-angle identity

as needed. The resulting integral can get messy for large m (or k).

If m is odd, then m = 2k + 1 for some integer k, and so

and then substitute u = cos(x) and du = -sin(x) dx, so that

Expand the binomial, and so on.

7

step-by-step explanation:

step-by-step explanation: