= Frullani integral
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{title2=$\int_0^\infty\frac{e^{-ax}-e^{-bx}}x\,dx=\log(b/a)$}
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For $0<a<b$, the nonnegative function $e^{-xy}\mathbf1_{\{x>0,\ a\le y\le b\}}$ is measurable and has integral $\int_a^bdy/y=\log(b/a)$. <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 $x=0$.
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