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The Gaussian integral

$$\intop_{-\infty}^{\infty} dx \exp(-x^2) = \sqrt{\pi} $$

is done in a very smart way. But where is the original document?

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    $\begingroup$ Gauss discovered neither the integral nor the "smart way" to evaluate it. $\endgroup$ – Alexandre Eremenko Nov 4 '18 at 15:10
  • $\begingroup$ You wrote it wrong - you need to indicate the upper and lower bounds. There is no indefinite integral. $\endgroup$ – Carl Witthoft Nov 5 '18 at 14:00
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At first I voted to close because Wikipedia clearly attributes that “smart way” to Poisson. But oddly, the linked document only quotes hearsay by Sturm (1859). The actual source is given by Bierens de Haan (1858, Table 40): Poisson’s Théorie mathématique de la chaleur (1835, §74).

(The integral was computed earlier by other methods — see loc. cit., Tables 36 & 37 and @KCd.)

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