Integrals of powers of the energy over the Bose-Einstein and Fermi-Dirac distributions are special cases of Lerch's zeta function. As a consequence, various results for these integrals follow directly from results recorded in the mathematical literature on special functions. The low-temperature series for the Fermi-Dirac case is considered as an example and derived from results recorded in the Bateman project. Alternative rigorous derivations of this series, useful for classroom presentation, are also given.