Integrating $\int \cos^3(x)\cos(2x) \, dx$

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.