  We evaluate two integrals over $x\in [0,1]$ involving products of the
function $\zeta_1(a,x)\equiv \zeta(a,x)-x^{-a}$ for $\Re (a)>1$, where
$\zeta(a,x)$ is the Hurwitz zeta function. The evaluation of these integrals
for the particular case of integer $a\geq 2$ is also presented. As an
application we calculate the $O(g)$ weak-coupling expansion coefficient
$c_{1}(\varepsilon)$ of the Casimir energy for a film with Dirichlet-Dirichlet
boundary conditions, first stated by Symanzik [Schr\"odinger representation and
Casimir effect in renormalizable quantum field theory, Nucl. Phys. B 190 (1981)
1-44] in the framework of $g\phi^4_{4-\varepsilon}$ theory.
