  In this paper we study the exponential functionals of the processes $X$ with
independent increments , namely $$I_t= \int _0^t\exp(-X_s)ds, _,\,\, t\geq 0,$$
and also $$I_{\infty}= \int _0^{\infty}\exp(-X_s)ds.$$ When $X$ is a
semi-martingale with absolutely continuous characteristics, we derive recurrent
integral equations for Mellin transform ${\bf E}( I_t^{\alpha})$,
$\alpha\in\mathbb{R}$, of the integral functional $I_t$. Then we apply these
recurrent formulas to calculate the moments. We present also the corresponding
results for the exponential functionals of Levy processes, which hold under
less restrictive conditions then in the paper of Bertoin, Yor (2005). In
particular, we obtain an explicit formula for the moments of $I_t$ and
$I_{\infty}$, and we precise the exact number of finite moments of
$I_{\infty}$.
