Calculus integration problem: $\int \sin^5 (x) \cos^2 (x)\,dx$ What's the integration of $$\int \sin^5 (x) \cos^2 (x)\,dx?$$

Using trig identities, you can show that:
$$\sin ^5(x) \cos ^2(x)=\frac{5 \sin (x)}{64}+\frac{1}{64} \sin (3
   x)-\frac{3}{64} \sin (5 x)+\frac{1}{64} \sin (7 x)$$
To do this, first use the "Power-reduction formulas" to reduce to get:
$$\sin^5(x)=\frac{10 \sin x - 5 \sin 3 x+ \sin 5 x}{16}$$
$$\cos^2(x)=\frac{1 + \cos (2 x)}{2}$$
And then use:
$$\cos (2 x) \sin (nx) = {{\sin((n+2)x) - \sin((n-2)x)} \over 2}$$
