We show that the class of L-constructible functions is closed
under integration for any P-minimal expansion of a p-adic field
(K,L). This generalizes results previously known for semi-algebraic
and sub-analytic structures. As part of the proof, we obtain a weak version of
cell decomposition and function preparation for P-minimal structures, a
result which is independent of the existence of Skolem functions. %The result
is obtained from weak versions of cell decomposition and function preparation
which we prove for general P-minimal structures. A direct corollary is that
Denef's results on the rationality of Poincar\'e series hold in any P-minimal
expansion of a p-adic field (K,L).Comment: 22 page