I was wondering what happens when evaluating an improper integral involving a trigonometric function where the denominator is a rational function with a zero at $x=0$. 

The example I have in mind is
$$\int\_{-\infty}^{\infty}\frac{sin(ax)}{x(x-i)(x+i)} dx$$

If I rewrite the sine in terms of the exponential and then evaluate two integrals, one for the upper half-plane and one for the lower I have a problem because of the singularity at $z=0$ which seems to be in both halves of the plane. Do I treat it in both integrals then? Any suggestions/comments are welcome.