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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 aretight as $x\rightarrow\infty$. We apply one of our inequalities to bound someexpressions involving this integral. Two of these expressions appear in Stein'smethod for variance-gamma approximation, and our bounds will allow for atechnical advancement to be made to the method.

Bounds for an integral of the modified Bessel function of the first kind and expressions involving it