  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
$\mathbb{R}^d$.
