  We construct Hrushovski-Kazhdan style motivic integration in certain
expansions of ACVF. Such an expansion is typically obtained by adding a full
section or a cross-section from the RV-sort into the VF-sort and some
(arbitrary) extra structure in the RV-sort. The construction of integration,
that is, the inverse of the lifting map L, is rather straightforward. What is a
bit surprising is that the kernel of L is still generated by one element,
exactly as in the case of integration in ACVF. The overall construction is more
or less parallel to the original Hrushovski-Kazhdan construction. As an
application, we show uniform rationality of Igusa zeta functions for
non-archimedean local fields with unbounded ramification degrees.
