Evaluate :

$\displaystyle \int_0^\infty \frac{x^2}{x^4+5x^2+4} dx$

$\displaystyle \int_0^\infty \frac{x^2}{(x^2+1)(x^2+4)} dx$

now I would like to extended this out into the complex plane and using the Cauchy integral to find the value, but not sure on how to do so exactly and what bounds to choose.

I figure I'll have something like:

$\displaystyle \int_C \frac{z^2}{(z^2+1)(z^2+4)} dx$