I'm trying to solve the following integral: $\int_{-\infty}^{+\infty} x^2e^{-x^2} dx$. I used integration by parts to get: $$-\frac{1}{2}[xe^{-x^2}]|_{-\infty}^{+\infty}+\int_{-\infty}^{+\infty}e^{-x^2} dx$$
I solved the gaussian integral easily, but I don't know how to evaluate the first term. After looking around a bit, I learned that it has to be 0. How? Thank you.