The theory of normal variance mixture distributions is used to provide elementary derivations of closed-form expressions for the definite integrals $\int_0^\infty x^{-2ν}\cos(bx)γ(ν,αx^2)\,\mathrm{d}x$ (for $ν>1/2$, $b>0$ $α>0$) and $\int_0^\infty x^{2ν-1}\cos(bx)Γ(-ν,αx^2)\,\mathrm{d}x$ (for $ν>0$, $b>0$ $α>0$), where $γ(a,x)$ and $Γ(a,x)$ 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.