The Gaussian integral is the improper integral defined as The function is known as the Gaussian function. Note how the graph takes the traditional bell-shape, the shape of the Laplace curve. You can use several methods to show that the integrand, the Gaussian function, has no indefinite integral that can be expressed in elementary terms. In other words, the integral resists the tools of elementary calculus. Still, there are several methods to evaluate it (differentiation under the integral sign, polar integration, contour integration, square of an integral, etc.) We will demonstrate the polar integration method.