  A classical result in the theory of Hopf algebras concerns the uniqueness and
existence of integrals: for an arbitrary Hopf algebra, the integral space has
dimension $\leq 1$, and for a finite dimensional Hopf algebra, this dimension
is exaclty one. We generalize these results to quasi-Hopf algebras and dual
quasi-Hopf algebras. In particular, it will follow that the bijectivity of the
antipode follows from the other axioms of a finite dimensional quasi-Hopf
algebra. We give a new version of the Fundamental Theorem for quasi-Hopf
algebras. We show that a dual quasi-Hopf algebra is co-Frobenius if and only if
it has a non-zero integral. In this case, the space of left or right integrals
has dimension one.
