  This is the second installment of a series of papers aimed at developing a
theory of Hrushovski-Kazhdan style motivic integration for certain types of
nonarchimedean $o$-minimal fields, namely power-bounded $T$-convex valued
fields, and closely related structures. The main result in the first
installment is a canonical isomorphism between the Grothendieck rings of
certain categories of definable sets, which is understood as a universal
additive invariant or a generalized Euler characteristic because the categories
do not carry volume forms. Here we introduce two types of volume forms into
each of the relevant categories, one takes values in the value group and the
other in the finer RV-sort. The resulting isomorphisms respect Jacobian
transformations --- that is, the change of variables formula holds --- and
hence are regarded as motivic integrals. As in the classical theory of
integration, one is often led to consider locally constant functions with
bounded support in various situations. For the space of such functions, the
construction may be fine-tuned so as to become more amenable to applications.
The modifications are nevertheless substantial and constitute the bulk of the
technical work.
