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.
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