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This is the beautiful Gaussian integral: \[I=\int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}.\]
It is also known as the Euler-Poisson integral. A cool way to prove the validity of the formula is considering $I^2$ and then doing a change of variables to polar coordinates.
The integral appears in probability (in relation to normal random variables) and physics (for example, in quantum mechanics).

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