I'm practising some residue calculations, and there's a few questions that require me to integrate from $-\infty$ to $\infty$. Usually this is fine, I do it over the big semicircle and let R go to infinity and sum the residues in the upper half plane. But in this question the $x$ term has a zero at zero.

$$\int^\infty\_{-\infty} \frac{sin(mx)}{x(a^2 +x^2)} dx \ \text{for} \ a>0 $$ 

I thought about using partial fractions but I can't get it to split. Can anyone offer help or a diagram of where to begin?

Thanks