## Gaussian integration of functions

Hi! I just studied the proof of the exactness of Gaussian integration on polynomials of degrees up to 2N-1. I was asking myself... is it possible to exactly integrate anything else using the same concept?
I mean, if I create the nonlinear system using a basis of a space of functions S, is it possible to exactly integrate, once weights and integration points are evaluated, any function in S?
Thanks!