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 $\int_0^\infty x^{-2\nu}\cos(bx)\gamma(\nu,\alpha x^2)\,\mathrm{d}x$ (for $\nu>1/2$, $b>0$ $\alpha>0$) and $\int_0^\infty x^{2\nu-1}\cos(bx)\Gamma(-\nu,\alpha x^2)\,\mathrm{d}x$ (for $\nu>0$, $b>0$ $\alpha>0$), where $\gamma(a,x)$ and $\Gamma(a,x)$ 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.