  Simple upper and lower bounds are obtained for the integral
$\int_0^x\mathrm{e}^{-\gamma t}t^\nu I_\nu(t)\,\mathrm{d}t$, $x>0$,
$\nu>-\frac{1}{2}$, $0<\gamma<1$. Most of our bounds for this integral are
tight as $x\rightarrow\infty$. We apply one of our inequalities to bound some
expressions involving this integral. Two of these expressions appear in Stein's
method for variance-gamma approximation, and our bounds will allow for a
technical advancement to be made to the method.
