Frullani integral (${\int }_0^{\infty }\frac{e^{-ax}-e^{-bx}}{x}dx=\log (b/a)$)

For $0<a<b$, the nonnegative function $e^{-xy}{1}_{\{x>0,a\le y\le b\}}$ is measurable and has integral ${\int }_a^{b}dy/y=\log (b/a)$. Tonelli theorem first proves integrability without assuming it; Fubini theorem then permits the order of integration to be exchanged. Integrating in $y$ gives the stated Frullani integral. The cancellation between the exponentials is essential at $x=0$.