  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 alternative
derivation 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 large
are also considered.
  A second section discusses the asymptotic expansion of an integral involving
a modified Bessel function that has recently been introduced as an extension of
the Whittaker function $M_{\kappa,\mu}(z)$. Numerical examples are provided to
illustrate the accuracy of the various expansions obtained.
