  We study integral theory for left (or right) Hopf left bialgebroids. Contrary
to Hopf algebroids, the latter ones don't necessary have an antipode $S$ but,
for any element $u$, the elements $u_{(1)} \otimes S(u_{(2)})$ (or
$u_{(2)}\otimes S^{-1}(u_{(1)})$ ) does exist. Our results extend those of G.
B\"ohm who studied integral theory for Hopf algebroids. We make use of recent
results about left Hopf left bialgebroids. We apply our results to the
restricted enveloping algebra of a restricted Lie Rinehart algebra.
