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

$I = \dfrac{2008^2}{\pi}$
Not sure how to bring graphs into answers, but here is a link to your function.

http://www.wolframalpha.com/input/?i=graph+%7Csin(pix)%7C

As you can see, the period is $1$, i.e. if

$f(x)=|\sin(\pi x)|$ then $f([0,1])=f([1,2])$.

So $\int_0^{2008}f(x) dx = 2008\int_0^1 f(x) dx$

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