A proof for the The Gaussian Integral

The definite integral of \(e^{-x^2}\) over the entire real line does have a beautiful result involving \(\pi\): $$\boxed{ \int_{-\infty}^{\infty} e^{-x^2}dx = \sqrt \pi}$$Consider the square of the integral: Let \(\displaystyle I=\int_{-\infty}^{\inf...