  We study integration over functions on superspaces. These functions are
invariant under a transformation which maps the whole superspace onto the part
of the superspace which only comprises purely commuting variables. We get a
compact expression for the differential operator with respect to the commuting
variables which results from Berezin integration over all Grassmann variables.
Also, we derive Cauchy--like integral theorems for invariant functions on
supervectors and symmetric supermatrices. This extends theorems partly derived
by other authors. As an physical application, we calculate the generating
function of the one--point correlation function in random matrix theory.
Furthermore, we give another derivation of supermatrix Bessel--functions for
U(k_1/k_2).
