How to integrate this conundrum?

[I<3Gauss]

Does anyone know how to prove the following statement? I haven't messed with integrals for awhile and I have to say that I am kind of rusty on this. From initial attempts, it seems the integral on the left is not something you can integrate directly... Maybe Taylor Expansion of cos^2(x) would help?

∫x/(1+cos^2(x)) = pi/(2*√2) (integrated from pi to 0)

Thanks guys!

 

[Whovian]

So you're asking for \displaystyle\int\limits_0^\pi\left(\dfrac{x\cdot dx}{1+\cos^2\left(x\right)}\right)? My first thought was substitute u=\dfrac\pi2-x and doing something like averaging out the results. Anyone?

EDIT: GRRR ... why isn't there a \mathrm on here?

 

[I<3Gauss]

Whovian, What gave you the idea to make such a substitution? What technique are you using?

 

[Whovian]

It's just a substitution that helps with a lot of trig integrals from 0 to pi/2.