Numerical quadrature of highly oscillatory integrals using derivatives

Numerical approximation of highly oscillatory functions is an area of research that has received considerable attention in recent years. Using asymptotic expansions as a point of departure, we derive Filon-type and Levin-type methods. These methods have the wonderful property that they improve with accuracy as the frequency of oscillations increases. A generalization of Levin-type methods to integrals over higher dimensional domains will also be presented