Numerical weighted integration of functions having mixed smoothness

We investigate the approximation of weighted integrals over $\mathbb{R}^d$ 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 $d \ge 2$, the upper bound is performed by sparse-grid quadratures with integration nodes on step hyperbolic crosses in the function domain $\mathbb{R}^d$.