Q:

Evaluate $\int_{0}^{\pi}\sin^5{\theta}\cos^2{\theta}\ d\theta$

I'm trying to find the mass of a spherical object with a given density function, and to do so I must solve this integral $$\int_{0}^{\pi}\sin^5{\theta}\cos^2{\theta}\ d\theta,$$
but no matter which method I choose (integration by parts, substitution, etc) I can't for the life of my figure out the anti-derivative.

A:

Let $x=\cos\theta$ then $dx=-\sin\theta d\theta$ hence
$$\int_0^\pi \sin^5\theta\cos^2\theta d\theta=\int_{-1}^1(1-x^2)^2x^2dx$$
Can you take it from here?

