I evaluated the Gaussian integral using polar substitution, and got that it is $\sqrt{\pi}$.
But my professor also asked us to compute the integral from $\int_{-\infty}^{\infty} e^{\frac{-x^2}{2}} dx$ and the integral from $\int_{0}^{\infty} x^2 e^{-x^2} dx$ using the evaluation of the Gaussian integral. How do I do that using my answer for the first part?
A nice trick is to compute $$f(t) = \int_{0}^{\infty} e^{-tx^2} dx = \frac{1}{2} \sqrt{\frac{\pi}{t}}$$ Using the same trick with polar coordinates.
Taking the derivative of Now just differentiate both sides and plug in $t=1$