TITLE: Integrating $\int \cos^3(x)\cos(2x) \, dx$
QUESTION [7 upvotes]: How would Integrate the following?
$$\int \cos^3(x)\cos(2x) \, dx.$$
I did 
$$\int \cos^3(x)(1-2\sin^2(x)) \, dx = 2\int \cos^3(x)-\cos^3x\sin^2x \, dx$$
But I find myself stuck....

REPLY [13 votes]: That's a good way to proceed. So our integral is
$$\int \cos^3 x(1-2\sin^2 x)\,dx.$$
Rewrite as
$$\int \cos x(1-\sin^2 x)(1-2\sin^2 x)\,dx$$
and let $u=\sin x$. We end up with
$$\int (1-u^2)(1-2u^2)\,du.$$
Expand and integrate.