The Gaussian is mathematically cooperative in almost every way except elementary integration.

The Gaussian integral is the famous definite integral of the bell-shaped function e^−x² across the entire real line, with the elegant value √π. Unlike many familiar integrals, it cannot be evaluated using an elementary antiderivative. Its classic solution takes a clever detour: square the integral, reinterpret it as a two-dimensional problem, and switch to polar coordinates. The result links exponential decay, geometry, and π in an unexpected way. It also provides the normalization behind the Gaussian, or normal, probability distribution. It is used primarily in probability theory, statistical physics, and signal processing.