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Proving $\int_{0}^{+\infty} e^{-x^2} dx = \frac{\sqrt \pi}{2}$

The primitive of $f(x) = \exp(-x^2)$ has no analytical expression, even so, it is possible to evaluate $\int f(x)$ along the whole real line with a few tricks. How can one show that $$ \int_{-\infty}^{\infty} \exp(-x^2) \,\mathrm{d}x = \sqrt{\pi} \space ? $$

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marked as duplicate by Ross Millikan, Eric Naslund, Américo Tavares, Nate Eldredge, J. M. May 13 '11 at 15:06

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

@Ross Millikan Damn, that one didn't show up in the related questions. It is quite a duplicate, the only difference is a predictable symmetry argument to double the answer. This question will probably be deleted. Oh, well. –  Luke May 13 '11 at 14:02
The related question searcher seems to be spotty. I am surprised both by what it finds and by what it misses. No problem. –  Ross Millikan May 13 '11 at 14:07

1 Answer 1

up vote 4 down vote accepted

Such an integral is called a Gaussian Integral

This link should help you out.

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