Fast expansion method for evaluating definite integrals with a variable upper limit and a composite or implicitly defined integrand

Given a continuous and sufficiently smooth composite or implicitly defined function on a bounded interval, it is shown that its definite integral with a variable upper limit can be approximately calculated at any point of the interval with high accuracy and minimum computer costs by applying an integrand representation based on fast sine expansions. Analytical quadrature rules are given, and a fast sine expansion algorithm consisting of simple easy-to-implement operations is described and illustrated by examples. The accuracy of the fast sine expansion method improves quickly as the number of retained Fourier series terms and the order of the boundary function are increased.