  A new highly accurate numerical approximation scheme based on a Gauss type
Clenshaw-Curtis Quadrature for Fredholm integral equations of the second kind,
whose kernel is either discontinuous or not smooth along the main diagonal, is
presented. This scheme is of spectral accuracy when the kernel is infinitely
differentiable away from the main diagonal, and is also applicable when the
kernel is singular along the boundary, and at isolated points on the main
diagonal. The corresponding composite rule is described. Application to
integro-differential Schroedinger equations with non-local potentials is given.
