  We prove a formula expressing the motivic integral (\cite{ls}) of a K3
surface over $\bC((t))$ with semi-stable reduction in terms of the associated
limit Hodge structure. Secondly, for every smooth variety over a
non-archimedean field we define an analogue of the monodromy pairing,
constructed by Grothendieck in the case of Abelian varieties, and prove that
our monodromy pairing is a birational invariant of the variety. Finally, we
propose a conjectural formula for the motivic integral of maximally degenerated
K3 surfaces over an arbitrary non-archimedean field and prove this conjecture
for Kummer K3 surfaces.
