it is very simple, let be u(x)= e^(-x^2), the, so u' (x)= -2xe^(-x^2), but we know that u' (x)= du (x)/dx, it implies du (x)/dx = -2xe^(-x^2), and
(-1/2)du (x) = xe^(-x^2)dx, (-1/2)integ.du (x)=integ.(xe^(-x^2))dx, finally
integ.(xe^(-x^2))dx = (-1/2)u(x)+ C= (-1/2)e^(-x^2) + C
