Integration & Exponential Function

I am trying to solve the integral, lower limit 0 and upper limit 1, x * e^(-x^2) but I am slightly confused.

Using a "u" substitution where u = -x^2, then du = -2xdx ... -du/2 = xdx.

So the integral becomes, e^u * -du/2.

I pull out the -1/2 constant in front of the integral and then integrate e^u * du.

This becomes -1/2 * (e^u). Then using the Fundamental Theorem, F(b) - F(a), I have: -1/2 * [ e^(-1^2) - e^(-0^2) ]

My answer is: -e/2 - 1/2

Could someone tell me if my answer is correct and if not, where I went wrong? Thanks!