  Let $(X_1,\ldots,X_n)$ be an i.i.d. sequence of random variables in
$\mathbb{R}^d$, $d\geq 1$. We show that, for any function $\varphi
:\mathbb{R}^d\rightarrow\mathbb{R}$, under regularity conditions, \[n^
{1/2}\Biggl(n^{-1}\sum_{i=1}^n\frac{\varphi(X_i)}{\widehat{f}^(X_i)}- \int
\varphi(x)\,dx\Biggr)\stackrel{\mathbb{P}}{\longrightarrow}0,\] where
$\widehat{f}$ is the classical 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 $\widehat{f}=f$. Although this paper
highlights some applications, we mainly address theoretical issues related to
the later result. We derive upper bounds for the rate of convergence in
probability. These bounds depend on the regularity of the functions $\varphi$
and $f$, the dimension $d$ and the bandwidth of the kernel estimator
$\widehat{f}$. Moreover, they are shown to be accurate since they are used as
renormalizing sequences in two central limit theorems each reflecting different
degrees of smoothness of $\varphi$. As an application to regression modelling
with random design, we provide the asymptotic normality of the estimation of
the linear functionals of a regression function. As a consequence of the above
result, the asymptotic variance does not depend on the regression function.
Finally, we debate the choice of the bandwidth for integral approximation and
we highlight the good behavior of our procedure through simulations.
