[SOLVED] Calculating an improper integral via complex analysis

I must calculate $\displaystyle \int _0 ^{\infty} \frac{dx}{x^4+1}$.

My attempt :

Let $\displaystyle f(z)=\frac{1}{z^4+1}$. I've calculated the poles of $\displaystyle f$ to be $\displaystyle z_i=\frac{\sqrt 2}{2}+i\frac{\sqrt2}{2}$, $\displaystyle \frac{\sqrt 2}{2}-i\frac{\sqrt2}{2}$, $\displaystyle -\frac{\sqrt 2}{2}+i\frac{\sqrt2}{2}$ and $\displaystyle -\frac{\sqrt 2}{2}-i\frac{\sqrt2}{2}$.

I chose the curve $\displaystyle \gamma _R$ as being the circle centered at $\displaystyle z=0$ with radius $\displaystyle R>1$.

So I have that $\displaystyle \int _{\gamma _R} f(z)dz=2 \pi i \sum _i Res(f,z_i)$.

It remains to calculate the residues of $\displaystyle f$, which I wasn't able to do.

I tried to express $\displaystyle f$ as a Laurent series but I didn't succeed in it.

I also tried to use the fact that the residues are worth $\displaystyle \lim _{z \to z_i} (z-z_i)f(z)$ but without success in evaluating the limit.

So how could I find the residues of $\displaystyle f$?

Thanks in advance... and sorry! I find all this very difficult to grasp in such a few time.