You might be better off trying to find the position, [;r;], as a function of time and then differentiating. To do this, use Gauss's Law to find the force acting on the particle as a function of [;r;] and then use Newton's Law: [F=ma=\frac{d^2r}{dt^2}] to get an ordinary differential equation...
Just evaluate the integral [;\int_{-\infty}^{\infty} dx;]. What do you get?
Also, [; <p>= m\frac{d<x>}{dt} ;], so if you can find [;<x>;], you should also be able to find [;<p>;].