I think the easiest way to integrate x e^(-x^2) is by substitution.
First left u = -x^2 then du = -2x dx then you can rearrange du=-2x dx to become
-(1/2)du=dx then the integration of
xe^(-x^2)dx becomes
e^u(-1/2)du which you can take out the constant of (-1/2)
results in [(-1/2)e^x^2] + C...