Numerical weighted integration of functions having mixed smoothness

We investigate the approximation of weighted integrals over Rd for integrands from weighted Sobolev spaces of mixed smoothness. We prove upper and lower bounds of the convergence rate of optimal quadratures with respect to n integration nodes for functions from these spaces. In the one-dimensional case (d=1), we obtain the right convergence rate of optimal quadratures. For d≥2, the upper bound is performed by sparse-grid quadratures with integration nodes on step hyperbolic crosses in the function domain Rd.