I want to compute $\displaystyle \int^{\infty}_{0}\frac{x}{x^4+1}dx$ using the residue theorem.
The poles in the upper half plane are:
location: $\large e^{\frac{\pi i}4}$, order: 1, residue: $\large\frac{1}{4}e^{\frac{3\pi i}2}$
location: $\large e^{\frac{3\pi i}4}$, order: 1, residue: $\large \frac{1}{4}e^{\frac{\pi i}2}$
The problem is that the integral from $-\infty$ to $\infty$ vanishes for symmetry reasons, so I cannot apply the standard approach of putting the half of a 1-sphere on top of the real axis and letting its radius go to infinity. If x was replaced with $x^2$ for instance, I could just divide the result by two. Is there another way of contour integration to evaluate the upper expression?