Semantic Insight

The Gaussian Integral, often called the Euler-Poisson integral, is the integral of the Gaussian function over the entire real line. It is fundamental to probability theory because it normalizes the normal distribution.

Intuitively, it calculates the area under the "bell curve." Even though the function has no elementary antiderivative, we can solve it by squaring the integral and moving to polar coordinates.

Symbolic Representation

Here is the standard derivation using polar coordinates:

$$ \int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi} $$

Proof step: $$ \left( \int{-\infty}^{\infty} e^{-x^2} \, dx \right)^2 = \int{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-(x^2+y^2)} \, dx \, dy $$

$$ = \int{0}^{2\pi} \int{0}^{\infty} e^{-r^2} r \, dr \, d\theta = \pi $$