On the approximate evaluation of oscillatory-singular integrals

In this paper an efficient numerical scheme is proposed for the numerical computation of the Cauchy type oscillatory integral ∫-11coswxxf(x)dx; where f(x) is a well-behaved function without having any kind of singularity in the range of integration [−1; 1]. The scheme is devised with the help of quadrature rule meant for the approximate evaluation of Cauchy principal value of integrals of the type ∫-11f(x)dx; and a quasi exact quadrature meant for the numerical integration of Filon-type integrals. The error bounds are determined and the scheme numerically verified by some standard test integrals.