I cant figure out where the singularities are on this Q!

Q: Use a large semicircular contour and the residue theorem to evaluate:

$\displaystyle \int_{-\infty}^{\infty} \frac{x^2}{x^4 + 1} dx$.

Lets call the integral above I.

Then I = $\displaystyle 2 \pi i \sum$ residues of $\displaystyle \frac{z^2}{z^4 + 1}$.

Then when i find these residues i sub them into $\displaystyle \frac{z^2}{4z^3} = \frac{1}{4z}$ to get the residues but i cant figure out what z should be. It SHOULD be $\displaystyle e^{\frac{\pi i}{4}}$ and $\displaystyle e^{\frac{3 \pi i}{4}}$ but i have no idea how to get those. I got $\displaystyle \sqrt{i}$ and $\displaystyle i \sqrt{i}$. Are those equivalent?!?