  If $f_1,f_2$ are smooth vector fields on an open subset of an Euclidean space
and $[f_1,f_2]$ is their Lie bracket, the asymptotic formula
$$\Psi_{[f_1,f_2]}(t_1,t_2)(x) - x =t_1t_2 [f_1,f_2](x) +o(t_1t_2),$$ where we
have set $ \Psi_{[f_1,f_2]}(t_1,t_2)(x) :=
\exp(-t_2f_2)\circ\exp(-t_1f_1)\circ\exp(t_2f_2)\circ\exp(t_1f_1)(x)$, is valid
for all $t_1,t_2$ small enough.
  In fact, the integral, exact formula \begin{equation}\label{abstractform}
\Psi_{[f_1,f_2]}(t_1,t_2)(x) - x =
\int_0^{t_1}\int_0^{t_2}[f_1,f_2]^{(s_2,s_1)} (\Psi(t_1,s_2)(x))ds_1\,ds_2 ,
\end{equation} where $ [f_1,f_2]^{(s_2,s_1)}(y) := D\Big(\exp(s_1f_1)\circ
\exp(s_2f_2{{)}}\Big)^{-1}\cdot [f_1,f_2](\exp(s_1f_1)\circ \exp(s_2f_2){(y)}),
$ with ${{y = \Psi(t_1,s_2)(x)}}$ has also been proven. Of course the integral
formula can be regarded as an improvement of the asymptotic formula. In this
paper we show that an integral representation holds true for any iterated
bracket made from elements of a family of vector fields ${f_1,\dots,f_{{k}}}$.
In perspective, these integral representations might lie at the basis for
extensions of asymptotic formulas involving nonsmooth vector fields.
