Gaussian Integral

Theorem
The Gaussian Integral is the integral over $\R$ of the function $f(x) = e^{-x^2}$.

Its value is $\sqrt{\pi}$.

That is:
 * $\displaystyle \int_{-\infty}^{+\infty} e^{-x^2} \ \mathrm d x = \sqrt \pi$

Proof
One of the most famous proofs of this result uses the trick of calculating instead a two-dimensional integral in polar coordinates, as follows.

Notice that, as $e^{-x^2 -y^2} = e^{-x^2} e^{-y^2}$:
 * $\displaystyle (1) \qquad \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} e^{-x^2 - y^2} \ \mathrm d x \ \mathrm d y = \left({\int_{-\infty}^{+\infty} e^{-x^2} \ \mathrm d x}\right)^2$

so if we calculate this two-dimensional integral, we will also have the value we want to find.

Changing to polar coordinates:

The result follows from $(1)$.