Can we use the residue theorem?
If you use that fact, then we have
But then changing the order of integration (which is justified because the original integral converges absolutely) doesn't really simply things.
The approach I had in mind was to use contour integration using a slightly unusual contour.
Performing the partial fraction expansion od the result (2) of my previous post we obtain...
The integrals in x can be solved in standard way with an approriate path in the complex plane obtaining...
Now is we substitute (4),(5), (6) and (7) in (3) we obtain the [incredible] result...
... so that is...
... and finally...
Let be the contour consisting of the line segment from the origin to (0,R); the arc from (R,0) to (0, IR); and the line segment from (0,IR) back to the origin.
has 4 simple poles at
The only pole inside of the contour is
So we have
(using L'Hospital's rule)
(ML inequality and triangle inequality)
Now we have
Now take the limit as R goes to infinity and equate the real portions on both sides to get