  We consider elliptic curves defined by an equation of the form
$y^2=x^3+f(t)$, where $f\in k[t]$ has coefficients in a perfect field $k$ of
characteristic not $2$ or $3$. By performing $2$ and $3$-descent, we obtain,
under suitable assumptions on the factorization of $f$, bounds for the number
of integral points on these curves. These bounds improve on a general result by
Hindry and Silverman. When $f$ has degree at most $6$, we give exact
expressions for the number of integral points of small height in terms of
certain subgroups of Picard groups of the $k$-curves corresponding to the $2$
and $3$-torsion of our curve. This allows us to recover explicit results by
Bremner, and gives new insight into Pillai's equation.
