I have this integral $I = \int_{0}^{+\infty} \frac{x\sin x}{(x^{2} + a^{2})^{2}} dx$. I tried to calculate it myself, but apparently I'm using the wrong residues formula, the answer doesn't come out.
$I = \int_{0}^{+\infty} f(z)dz =\int_{0}^{+\infty} \frac{z\sin z}{(z^{2} + a^{2})^{2}} dz$. Poles: $z^{2} + a^{2} = 0 \to z_{1,2} = \pm ia$ - poles are order $n =4.$
The residue of 4th order is: $\text{Res} f(z_{1,2}) = \lim_{z\to z_{1,2}}(\frac{1}{(3)!})\frac{d^{3}}{dz^{3}}[(z-z_{1,2})^{4}f(z)]$ if I put the rest of the data to that formula I won't get the right answer ( which is $I = \frac{π e^{-a}}{4a}$).
What could be wrong?