Residue Theorem Application for Integrating Functions with Poles of Higher Order

[zetafunction]

Homework Statement

Calculate the integral $$I(k)= \int_{-\infty}^{\infty} \frac{dx}{(x^{2}+1)^{k}}$$ with 'k' being a real number

Homework Equations

the integral equation above

The Attempt at a Solution

from the residue theorem , there is a pole of order one at $$2+ix=0$$ , my problem is the pole of order 'k' at s=i and s=-i , in order to handel with this pole i have thought that using residue theorem

$$\frac{1}{\Gamma(s)}D^{k-1}((s-i)^{k}\frac{1}{(x^{2}+1)^{k}}$$

evaluated at both s=i and s=-i

 

[Nino MMP]

.the problem is that in my opinion i can calculate the residue at s=i and s=-i where the order of the pole is 'k' , but the problem is how to calculate the integral I(k) .any advice ? thanks