Frullani integral

Remarks

For $0 < \varepsilon < R$, the substitutions $t = ax$ and $t = bx$ give $$\int_\varepsilon^R \frac{f(ax) - f(bx)}{x}\, dx = \int_{a\varepsilon}^{b\varepsilon} \frac{f(t)}{t}\, dt - \int_{aR}^{bR} \frac{f(t)}{t}\, dt.$$ As $\varepsilon \to 0$ the first integral tends to $f(0) \ln(b/a)$ by continuity at $0$, and as $R \to \infty$ the second tends to $f(\infty) \ln(b/a)$. Example: with $f(x) = e^{-x}$ one gets $\int_0^\infty \frac{e^{-ax} - e^{-bx}}{x}\, dx = \ln \frac{b}{a}$. The limit at infinity matters: for $f(x) = \sin x$ the right side has no meaning, although the integral still converges.

Going deeper

For a student

Frullani integrals appear in the computation of integral representations of logarithms, such as $\ln(b/a) = \int_0^\infty \frac{e^{-ax} - e^{-bx}}{x}\, dx$. The proof is a change of variables in an improper integral together with continuity at the two ends. Variants replace continuity by local integrability with suitable mean values of $f$ near $0$ and near infinity.