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 $ \int_{ - 1}^{1} {\frac{\cos wx}{x}} f( x ){\text{d}}x $; 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 $ \int_{ - 1}^{1} {\frac{f( x )}{\text{d}x}} $; 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