I need to evaluate the following integral using complex analysis methods $\int \limits_{-\infty}^{\infty}\frac{1}{1+x^4}dx$.
$f(z)=\frac{1}{1+z^4}=\frac{1}{(z^2+i)(z^2-i)}=\frac{1}{(z+\sqrt{i})(z-\sqrt{i})(z+\sqrt{-i})(z-\sqrt{-i})}$
We know that $f(z)$ is analytic on the whole complex plane except from the 4 poles. These are 4 simple poles which correspond to $z=\pm\sqrt{i}$ and $z=\pm\sqrt{-i}$.
This is the stage where I would like verification of the step. I no longer considered the $z=-\sqrt{i}$ and the $z=+\sqrt{-i}$ poles as they dont fall within the semicircular contour typically used when calculating these problems.
I proceeded with the calculations of the remaining two poles to find their residues and found that they both have a residue of $\frac{1}{(4\sqrt{i})}$.
Finally, using the residue theorem, I added the two residues together to obtain $\frac{2}{(4\sqrt{i})}$and multiplied by $2\pi$i to give $\pi\sqrt{i}$.
Can anybody confirm or verify that this is the correct evaluation of the integral or perhaps where I may have gone astray?
