How do I prove that the integral from negative infinty to infinty of $e^{-.5x^2}$ is $\sqrt{2\pi}$?

Question

I've tried integration by parts, u-substitution, and a Taylor Series expansion. I feel like the gamma function would play a role in this proof, but I'm not sure how. Thank you in advance.

Answer 1

Your integral looks like a Gaussian integral. The answer to your question is $$I = \sqrt{2\pi}$$ More generally, $$\int_{-\infty}^{\infty} e^{-a(x+b)^2}dx = \sqrt{\frac \pi a}$$ You might want to check out the linked wiki article for a proof/more details.