$\int_{0}^{2008}x|\sin\pi x| dx$

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:

$I = \dfrac{2008^2}{\pi}$