  We construct a new motivic integration morphism, the so-call bounded
integral, that interpolates both the integration morphisms with and without
volume forms of Hrushovski and Kazhdan. This is done within the framework of
model theory of algebraically closed valued fields of equicharacteristic zero.
As an application, we recover and extend some results of Hrushovski and Loeser
about the motivic Milnor fiber.
