Quadrature rules using first derivatives for oscillatory integrands

We consider the integral of a function y(x), I(y(x))= ∫ −1 1 y(x) dx and its approximation by a quadrature rule of the form Q N(y(x))= ∑ k=1 N w ky(x k)+ ∑ k=1 N α ky′(x k), i.e., by a rule which uses the values of both y and its derivative at nodes of the quadrature rule. We examine the cases when the integrand is either a smooth function or an ω dependent function of the form y(x)=f 1(x) sin(ωx)+f 2(x) cos(ωx) with smoothly varying f 1 and f 2. In the latter case, the weights w k and α k are ω dependent. We establish some general properties of the weights and present some numerical illustrations.