  The theory of integrals is used to analyse the structure of Hopf algebroids,
introduced in math.QA/0302325. We prove that the total algebra of the Hopf
algebroid is a separable extension of the base algebra if and only if it is a
semi-simple extension and if and only if the Hopf algebroid possesses a
normalized integral. The total algebra of a finitely generated and projective
Hopf algebroid is a Frobenius extension of the base algebrahe Hopf algebroid
possesses anormalized integral. We give also a sufficient and necessary
condition in terms of integrals under which it is a quasi-Frobenius extension
and illustrate by an example that this condition does not hold true in general.
Our results are generalizations of classical results on Hopf algebras.
