A probabilistic proof of some integral formulas involving incomplete gamma functions

The theory of normal variance mixture distributions is used to provide elementary derivations of closed-form expressions for the definite integrals ${\int }_0^{\mathrm{\infty }}{x}^{-2\nu }\cos (bx)\gamma (\nu ,\alpha x^2)\mathrm{d}x$ (for $\nu >1\mathrm{/}2$, $b>0$ $\alpha >0$) and ${\int }_0^{\mathrm{\infty }}{x}^{2\nu -1}\cos (bx)\mathrm{\Gamma }(-\nu ,\alpha x^2)\mathrm{d}x$ (for $\nu >0$, $b>0$ $\alpha >0$), where $\gamma (a,x)$ and $\mathrm{\Gamma }(a,x)$ are the lower and upper incomplete gamma functions, respectively. The method of proof is of independent interest and could be used to derive further new definite integral formulas.