Thursday, July 5, 2018

calculus - Simplest way to integrate this expression : $int_{-infty}^{+infty} e^{-x^2/2} dx$




I'm toying around with statistics and calculus for a project of mine and I'm trying to find the simplest/fastest way to integrate this formula :



$$\int_{-\infty}^{+\infty} e^{-x^2/2} dx$$




  • I do not want to use a table.

  • I'm taking this opportunity to get more practice with my new calculus skills

  • It seems that a Taylor series approx is the only way to go




Best Regards


Answer



If we set $$I := \int_{\mathbb{R}} \exp \left(- \frac{x^2}{2} \right) \, dx,$$
then



$$I^2 = \int_{\mathbb{R}} \int_{\mathbb{R}} \exp \left( - \frac{x^2+y^2}{2} \right) \, dx \, dy.$$



Introducting polar coordinates, i.e.




$$\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} r \cos \varphi \\ r \sin \varphi \end{pmatrix},$$



yields



$$I^2 = \int_{r=0}^{\infty} \int_{\varphi=0}^{2\pi} e^{-r^2/2} r \, dr \, d\varphi = \left( \int_{0}^{\infty} r e^{-r^2/2} \, dr \right) \left( \int_{\varphi=0}^{2\pi} d \varphi \right).$$



This expression can be easily calculated.


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