Hi. We can rely on the Residue Theorem to show this by considering a closed contour with indentations around and over each pole along the real axis and then circling back about the upper half-plane. We then have:
(since we're only going around at each pole and in the reverse direction). Then:
where are the residues at the poles. I'll leave it to you to show that the sum of those three residues is zero and also should say something about how the integral over the remaining large semi-circle of the contour tends to zero as .