TITLE: $\int_{0}^{2008}x|\sin\pi x| dx$
QUESTION [6 upvotes]: Evaluate: 
$$\int_{0}^{2008}x|\sin\pi x| dx$$
That modulus sign is causing problems. How do I handle it?
I am trying integration by parts
I have even evaluated: $\int_0^1 {|\sin \pi x|}= \frac 2 \pi$. Not sure how to utilise it in the problem. 
I just need help with the modulus part.

REPLY [8 votes]: Let $I$ denote the given integral. 
Using, 
$\int_{a}^b f(a+b-x)dx = \int_{a}^{b} f(x)dx$
we get: 
$2I = \int_0^{2008} 2008 |\sin \pi x|dx$
Using the periodicity of $|\sin \pi x|$, we obtain: 
$I = \dfrac{2008^2}{\pi}$