Integration of Gaussian function

Integration by substitution

$$\begin{array}{rl}{\int }_{-\infty }^{+\infty }{e}^{-x^2}dx=& \sqrt{{\int }_{-\infty }^{+\infty }{e}^{-s^2}ds\int e^{-t^2}dt}\\ =& \sqrt{{\int }_{-\infty }^{+\infty }{e}^{-(s^2+t^2)}dsdt}\\ =& \sqrt{{\int }_0^{2\pi }{\int }_0^{+\infty }{e}^{-r^2}rdrd\theta }\\ =& \sqrt{{\int }_0^{2\pi }{\int }_0^{+\infty }{e}^{-r^2}rdrd\theta }\\ =& \sqrt{\pi }\end{array}$$