---
tag: 0DFT1GP1JP
kind: example
name: Gaussian integral
aliases: [euler-poisson-integral]
msc: [26A42, 40A10]
instance_of: [05KQ3F99BQ]
uses: [0VCTBJMEJF, 0FR1BDKK10, 0RMVS7RXZX]
sources:
  - url: https://en.wikipedia.org/wiki/Gaussian_integral
    locator: lead section
same_as:
  wikidata: Q1060321
  wikipedia: Gaussian integral
---

## Example

The improper integral of $e^{-x^2}$ over $\mathbb{R}$ converges, and
$$\int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}.$$
Convergence follows from the [[comparison test|0FR1BDKK10]] with $e^{-|x|}$ for $|x| \ge 1$.
For the value, let $I$ be the integral. In polar coordinates, $I^2 = \int_0^{2\pi} \int_0^\infty e^{-r^2} r \, dr \, d\theta = \pi$.

## Remarks

The function $e^{-x^2}$ has no elementary antiderivative. So the second fundamental theorem does not give the value.
A change of variables gives $\int_{-\infty}^{\infty} e^{-a x^2} \, dx = \sqrt{\pi / a}$ for $a > 0$. Mathlib states this as `integral_gaussian`.
Abraham de Moivre found this type of integral in 1733. Carl Friedrich Gauss published the exact value in 1809.

## Explanations

### L1 human

The bell curve $e^{-x^2}$ has total area $\sqrt{\pi}$.
This number gives the constant in the normal distribution of probability theory.