I would like to compute the Gaussian integral $\int_{-\infty}^{+\infty}e^{-x^2/2} dx$ using a polar substitution.
I know the usual proof using Fubini's theorem and the polar substitution of variables.
However, I do not want to use the formula ${\displaystyle \int _{\varphi (U)}f(\mathbf {v} )\,d\mathbf {v} =\int _{U}f(\varphi (\mathbf {u} ))\left|\det(D\varphi )(\mathbf {u} )\right|\,d\mathbf {u}}$ but rather re-demonstrate it in this precise case (that is, polar substitution).