In this paper, motivated by the contributions of several researchers, including (for example) Riemann, Kronecker, Lerch, Ramanujan, and Glaisher on some special integrals known as Mordell’s integrals, we consider the problem of the evaluation of some generalized Mordell integrals by applying the Mellin-Barnes-type contour integration. In particular, we have evaluated the following integrals: I1:=∫0∞e−at2+btect+gdtandI2:=∫−∞∞e−at2+btect+gdt for real or complex parameters a,b,c, and g with ℜ(a)>0 in terms of Meijer’s G-function and the generalized hypergeometric function pFq.