Gaussian integral

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This article/section deals with mathematical concepts appropriate for late high school or early college.

The Gaussian integral is the integral:

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It has a value of $\text{[math]}$ . The value is needed to normalize the Normal distribution.

Derivation

First look at the double integral

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Separating it,

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So, the double integral is merely the square of the Gaussian integral.

Now, do the double integral in polar co-ordinates. $\text{[math]}$ and $\text{[math]}$ , so:

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Substituting $\text{[math]}$ into the integral,

$\text{[math]}$

Therefore,

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