  We consider a perturbed KdV equation:
  [\dot{u}+u_{xxx} - 6uu_x = \epsilon f(x,u(\cdot)), \quad x\in \mathbb{T},
\quad\int_\mathbb{T} u dx=0.]
  For any periodic function $u(x)$, let
$I(u)=(I_1(u),I_2(u),...)\in\mathbb{R}_+^{\infty}$ be the vector, formed by the
KdV integrals of motion, calculated for the potential $u(x)$. Assuming that the
perturbation $\epsilon f(x,u(\cdot))$ is a smoothing mapping (e.g. it is a
smooth function $\epsilon f(x)$, independent from $u$), and that solutions of
the perturbed equation satisfy some mild a-priori assumptions, we prove that
for solutions $u(t,x)$ with typical initial data and for $0\leqslant t\lesssim
\epsilon^{-1}$, the vector $I(u(t))$ may be well approximated by a solution of
the averaged equation.
