  We show that the class of $\mathcal{L}$-constructible functions is closed
under integration for any $P$-minimal expansion of a $p$-adic field
$(K,\mathcal{L})$. 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 $(K,\mathcal{L})$.
