  Let $(X_1,\ldots ,X_n)$ be an i.i.d. sequence of random variables in $\R^d$,
$d\geq 1$, for some function $\varphi:\R^d\r \R$, under regularity conditions,
we show that \begin{align*} n^{1/2} \left(n^{-1} \sum_{i=1}^n
\frac{\varphi(X_i)}{\w f^{(i)}(X_i)}-\int_{} \varphi(x)dx \right)
\overset{\P}{\lr} 0, \end{align*} where $\w f^{(i)}$ is the classical
leave-one-out kernel estimator of the density of $X_1$. This result is striking
because it speeds up traditional rates, in root $n$, derived from the central
limit theorem when $\w f^{(i)}=f$. As a consequence, it improves the classical
Monte Carlo procedure for integral approximation. The paper mainly addressed
with theoretical issues related to the later result (rates of convergence,
bandwidth choice, regularity of $\varphi$) but also interests some statistical
applications dealing with random design regression. In particular, we provide
the asymptotic normality of the estimation of the linear functionals of a
regression function on which the only requirement is the H\"older regularity.
This leads us to a new version of the \textit{average derivative estimator}
introduced by H\"ardle and Stoker in \cite{hardle1989} which allows for
\textit{dimension reduction} by estimating the \textit{index space} of a
regression.
