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  "body": "\n## Example\n\nThe improper integral of $e^{-x^2}$ over $\\mathbb{R}$ converges, and\n$$\\int_{-\\infty}^{\\infty} e^{-x^2} \\, dx = \\sqrt{\\pi}.$$\nConvergence follows from the [[comparison test|0FR1BDKK10]] with $e^{-|x|}$ for $|x| \\ge 1$.\nFor the value, let $I$ be the integral. In polar coordinates, $I^2 = \\int_0^{2\\pi} \\int_0^\\infty e^{-r^2} r \\, dr \\, d\\theta = \\pi$.\n\n## Remarks\n\nThe function $e^{-x^2}$ has no elementary antiderivative. So the second fundamental theorem does not give the value.\nA change of variables gives $\\int_{-\\infty}^{\\infty} e^{-a x^2} \\, dx = \\sqrt{\\pi / a}$ for $a \u003e 0$. Mathlib states this as `integral_gaussian`.\nAbraham de Moivre found this type of integral in 1733. Carl Friedrich Gauss published the exact value in 1809.\n\n## Explanations\n\n### L1 human\n\nThe bell curve $e^{-x^2}$ has total area $\\sqrt{\\pi}$.\nThis number gives the constant in the normal distribution of probability theory.\n",
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