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We consider the asymptotic expansion of Kr\"atzel's integral\[F_{p,\nu}(x)=\int_0^\infty t^{\nu-1} e^{-t^p-x/t}\,dt\qquad(|\arg\,x|<\pi/2),\] for $p>0$ as $|x|\to \infty$ in the sector$|\arg\,x|<\pi/2$ employing the method of steepest descents. An alternativederivation of this expansion is given using a Mellin-Barnes integral approach.The cases $p<0$, $\Re (\nu)<0$ and when $x$ and $\nu$ ($p>0$) are both largeare also considered.  A second section discusses the asymptotic expansion of an integral involvinga modified Bessel function that has recently been introduced as an extension ofthe Whittaker function $M_{\kappa,\mu}(z)$. Numerical examples are provided toillustrate the accuracy of the various expansions obtained.

The asymptotic expansion of Kratzel's integral and an integral related to an extension of the Whittaker function