How about showing us HOW you got that result? I think I would be inclined to integrate around a loop in the complex plane- integrate from -R to R on the real line, then counter-clockwise around the half circle in the positive i half plane with radius R. The function has a pole at z= i. (And a removable discontinuity at z= 0.) Then, of course, take the limit as R goes to infinity. Since the integrand obviously goes to 0 as |z| goes to infinity, the integral depends completely on the residue at z= 1. Did you calculate the residue?
But that might be exactly what you are trying to do or it might be completely different. What, if anything, you did wrong, we can't say until we see what you did.
And since this does not have anything, directly, to do with "differential equations", I am moving it to "Calculus and Analysis".
