Numerical integration is not really a field that you expect to have surprising theorems. Yet, the existence of Gaussian quadrature is in of itself surprising. In the elementary methods that you learn in calculus (such as midpoint rule or Simpson’s rule), you evaluate the function at regularly spaced nodes. A more effective technique is to choose so that you can integrate polynomials up to some degree exactly. The best choice? The roots of an orthogonal polynomial.

The proofs are elementary, and can be found in this note by John Cook.