  The Hurwitz-type Euler zeta function is defined as a deformation of the
Hurwitz zeta function: \begin{equation*}
\zeta_E(s,x)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+x)^s}. \end{equation*} In this
paper, by using the method of Fourier expansions, we shall evaluate several
integrals with integrands involving Hurwitz-type Euler zeta functions
$\zeta_E(s,x)$. Furthermore, the relations between the values of a class of the
Hurwitz-type (or Lerch-type) Euler zeta functions at rational arguments have
also been given.
