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 ${\mathbb{R}}^d$.