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 R∞ 0 x−2ν cos(bx)γ(ν, αx2) dx (for ν > 1/2, b > 0 α > 0) and R∞0 x2ν−1 cos(bx)Γ(−ν, αx2) dx (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.