  M.F. Singer [Liouvillian first integrals of differential equations, Trans.
Amer. Math. Soc. 333 (1992), 673--688] proved the equivalence between
Liouvillian integrability and Darboux integrability for two dimensional
polynomial differential systems. In this paper we will extend Singer's result
to any finite dimensional polynomial differential systems. We prove that if an
n--dimensional polynomial differential system has n-1 functionally independent
Darboux Jacobian multiplier then it has n-1 functionally independent
Liouvillian first integrals. Conversely if the system is Liouvillian integrable
then it has a Darboux Jacobian multiplier.
