Gaussian Integral

Integration of Gaussian Function $e^{-x^2}$, a.k.a. Euler-Poisson integral

it’s a constant $\sqrt{\pi }$, you can get this by Integration by substitution

$$\begin{array}{rl}\int \_{-\infty }^{+\infty }{e}^{-x^2}dx\int \_{-\infty }^{+\infty }{e}^{-y^2}dy& =\int \_{-\infty }^{+\infty }\int \_{-\infty }^{+\infty }{e}^{-(x^2+y^2)},dx,dy\\ & =\int \_0^{2\pi }\int \_0^{+\infty }{e}^{-r^2}r,dr,d\theta \\ & =\int \_0^{2\pi }(-\frac{1}{2})\int \_0^{+\infty }{e}^{-r^2}d(-r^2)d\theta \\ & =\int \_0^{2\pi }(-\frac{1}{2})(e^{-r^2}{|}_0^{+\infty })d\theta \\ & =\pi \\ ⟹{\int }_{-\infty }^{+\infty }{e}^{-x^2}dx& =\sqrt{\pi }\end{array}$$