  Let $G$ be a connected semisimple Lie group with a finite center, with a
maximal compact subgroup $K$. There are two general methods to construct
representations of $G$, namely, ordinary parabolic induction and cohomological
parabolic induction. We define Eisenstein integrals relative to cohomological
inductions which generalize Flensted-Jensen's fundamental functions for
discrete series. They are analogous to Harish-Chandra's Eisenstein integrals
related to ordinary inductions. We introduce the notion of Li positivity of a
$K$ type in a representation of $G$ which is extremely useful in the study of
branching laws. As an application of our integral, we show that the minimal $K$
types of many interesting representations are Li positive. These include all
irreducible unitary representations with nonzero cohomology.
