I have the following integral, which I dont know how to deal with:
$\int_0^\infty J_0(ax)J_0(bx)e^{-x^2}xdx.$
Integration by parts doesnt help.
I'm sure that integral must result in something like
$2\pi I_0(2ab)e^{-(a^2+b^2)},$
but I dont know how to prove it.