The derivative of \(\displaystyle e^{-x^2}\) is \(\displaystyle -2xe^{-x^2}\)

Therefore an antiderivative of \(\displaystyle -2xe^{-x^2}\) is \(\displaystyle e^{-x^2}\)

And an antiderivative of \(\displaystyle xe^{-x^2}\) is \(\displaystyle -\frac12 e^{-x^2}\)