back to physics

Ok, I need to do some integrals. I'm trying to find the number of electrons scattering per unit time into a given longitudinal velocity.
I need to do the integral:

where is typically a Gaussian.
I can do the integral first which seems like the right thing to do. But then to check it, I can integrate over from the momentum acceptance to infinity and get the Touschek lifetime. But I can't see how it gives the right answer. Anyway, its fun to get back to this stuff.