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I have the integral ##\int_{-\infty}^{\infty} x^2 e^{-x^2} ~dx##. Is there any simple way to integrate this, given that that I already know that the value of the Gaussian integral is ##\sqrt{\pi}##?
What I found was that ##\int_{-\infty}^{\infty}e^{-x^2} ~dx = 2\int_{-\infty}^{\infty}x^2 e^{-x^2} ~dx##, so by original integral is ##\sqrt{\pi} / 2##. To do this though I had to start from the original Gaussian integral and integrate by parts to get the integral that I have. Is that trick of integrating by parts the original Gaussian integral to get what I want what you had in mind?Try integration by parts.
Is that trick of integrating by parts the original Gaussian integral to get what I want what you had in mind?