  The theme of the present paper is numerical integration of $C^r$ functions
using randomized methods. We consider variance reduction methods that consist
in two steps. First the initial interval is partitioned into subintervals and
the integrand is approximated by a piecewise polynomial interpolant that is
based on the obtained partition. Then a randomized approximation is applied on
the difference of the integrand and its interpolant. The final approximation of
the integral is the sum of both. The optimal convergence rate is already
achieved by uniform (nonadaptive) partition plus the crude Monte Carlo;
however, special adaptive techniques can substantially lower the asymptotic
factor depending on the integrand. The improvement can be huge in comparison to
the nonadaptive method, especially for functions with rapidly varying $r$th
derivatives, which has serious implications for practical computations. In
addition, the proposed adaptive methods are easily implementable and can be
well used for automatic integration.
