I need some help solving this integral, seems nasty to me!
$$
\int e^{-x}\cos4x\cos2x\,\mathrm dx
$$
I tried integration by parts, but that seemed to me of no use.
There's also a similar one, maybe it might help solving this one or vice versa.
$$
\int x\sin x\sin 2x\sin 3x\,\mathrm dx
$$
Again, please try to give just hints…! (As I always ask :D)
Best Answer
Use the fact that
$$2 \cos{a x} \cos{b x} = \cos{(a-b) x} + \cos{(a+b) x}$$
and
$$\int dx\: e^{p x} \cos{q x} = \frac{p \cos{q x} + q \sin{q x}}{p^2+q^2} e^{p x} + C$$
where $C$ is a constant of integration.