2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/161816We introduce a direct image formalism for constructible motivic functions. One deduces a very general version of motivic integration for which a change of variables theorem is proved. These constructions are generalized to the relative framework, in which we develop a relative version of motivic integration. These results have been announced in math.AG/0403349 and math.AG/0403350. Main results and statements unchanged. Many minor slips corrected and some details added.Final versionAlgebraic GeometryLogicNumber Theory14G20 (Primary) 03C10, 11S80, 11U09, 15G05 (Secondary)text