Integral of x^2/(x^6+1) from 0 to infinity using residues

[gtfitzpatrick]

Homework Statement

show $$^{\infty}_{0}\int$$ x² / (x⁶ + 1) = $$\pi$$/6

Homework Equations

The Attempt at a Solution

= 1/2 $$\int^{\infty}_{-\infty}$$ ...

I am not sure how to go about this any pointers pls?

 

[lanedance]

you're really supposed to attempt some working

but note the function is symmetric on the real axis... and the magnitude will get small as the radius gets big, this should help you pick a contour to integrate

it will help to find the poles as well...

 

[Count Iblis]

Note that you don't need to integrate over the half circle and come back along the negative real axis. You would then have to sum over the three poles in the upper half plane. Instead, you can return along the line z = r exp(i pi/3).