TITLE: Faster way to compute the integral
QUESTION [5 upvotes]: Here it is:
$$\int_o^\pi x\cos^4x\,dx$$
I used integration by parts but I would be grateful if someone told me an alternate method to compute the integral faster.

REPLY [6 votes]: The well known formula
$$\int_0^\pi xf(\sin x )dx=\frac \pi2\int_0^\pi f(\sin x )dx$$
yields
$$I=\int_{o}^{\pi} x\cos^4x=\frac \pi2\int_0^\pi \cos^4xdx=\pi\int_0^{\pi/2} \cos^4xdx$$
then let $x=\pi/2-y$ and have that 
$$I=\pi\int_0^{\pi/2} \cos^4xdx=\pi\int_0^{\pi/2} \sin^4xdx$$
$$2I=\pi\int_0^{\pi/2} (\sin^4x+\cos^4x)dx=\frac \pi4\int_0^{\pi/2} (\cos(4 x)+3) dx= \frac{3 \pi^2}{8}$$
$$I= \frac{3 \pi^2}{16}.$$
Chris