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