  By applying the inverse Mellin transform to some simple closed form
identities, a number of relationships are established that connect integrals
containing Riemann's and Hurwitz' zeta functions and their alternating
equivalents. Interesting special cases involving improper integrals containing
$\zeta(\sigma+it)$ and a minimum of other functions in the integrand are
identified. Many of these integrals do not appear in the literature and can be
verified numerically. In one limit, the use of analytic continuation generates
a family of improper integrals containing only the real and imaginary parts of
$\zeta(\sigma+it)$ with and without simple trigonometric factors; the
associated closed form contains an (unclassified) entity that has many of the
attributes of an essential singularity, but probably is not. Consequently, this
means that the associated integrals are indeterminate (i.e. non-single valued),
so a new symbol is introduced to label the indeterminism. Much of this paper
studies this singularity from several angles in an attempt to resolve the
associated ambiguities. This is done by establishing a self-consistent way to
remove the singularity and thereby evaluate members of the family. These new
closed-form identities provide insight into the value of integrals of general
interest. Some implications are proposed.
