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

How does one integrate $\int e^{-x^2}\,dx$? I read somewhere to use polar coordinates.

How is this done? What is the easiest way?

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marked as duplicate by Mike Spivey, t.b., Américo Tavares, Ross Millikan, Willie Wong Apr 23 '11 at 16:13

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.

As an indefinite integral, this is not expressible by elementary functions. I guess you mean a definite integral? –  Raeder Apr 23 '11 at 12:00

3 Answers 3

You can integrate the function $e^{-x^2}$ only as a definite integral, as you mention you can do it using polar coordinates as follows:

Let $I = \int_{-\infty}^{\infty} e^{-x^2}\,dx$ then multiply it by $I = \int_{-\infty}^{\infty} e^{-y^2}\,dy$ so we have $I^2 = \left(\int_{-\infty}^{\infty} e^{-x^2}\,dx\right) \left(\int_{-\infty}^{\infty} e^{-y^2}\,dy\right) = \left(\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-(x^2 + y^2)}\,dx\,dy\right)$ now we can change this integral to polar coordinates using $ \rho^2 = x^2 + y^2$ and now $ I^2 = \int_{0}^{2\pi} \int_{-\infty}^{\infty} e^{\rho^2} \rho \, d\rho \, d\theta = \pi$ and finally $I = \int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}$.

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We had a nice discussion of different ways to prove this over on Tim Gowers' blog a few years ago.

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The function you have mentioned is called the gaussian integral.... Read more about it on wiki. It has explanation regarding solving it by using the polar co-ordinates too ..............

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