Computing the integral $\int_{-\infty}^{+\infty} \frac{\cos x}{(x^2+a^2)^2}\,dx$ using complex analysis

Question

I want to compute the integral $\displaystyle\int_{-\infty}^{+\infty} \dfrac{\cos x}{(x^2+a^2)^2}~dx$ for $a>0$

How does one do so. Iwas thinking maybe I have to use the theorem of residues that states that: $$\int_{\gamma} f(z)~dz = 2 \pi i \sum_{i=-1}^n Res(f,c_i)$$

Answer 1

Use that

$$f(z) = \frac{e^{iz}}{(z^2+a^2)^2}$$

and integrate around half-circle in the upper half plane.