On numerical computation of integrals with integrands of the form f(x)sin(w/xr) on [0, 1]

With existing numerical integration methods and algorithms it is difficult in general to obtain accurate approximations to integrals of the form ∫01f(x)sin(ωxr)dxor∫01f(x)cos(ωxr)dx,(r>0) where f is a sufficiently smooth function on [0, 1]. Gautschi has developed software (as scripts in Matlab) for computing these integrals for the special case r=ω=1. In this paper, an algorithm (as a Mathematica program) is developed for computing these integrals to arbitrary precision for any given values of the parameters in a certain range. Numerical examples are given of testing the performance of the algorithm/program.