Frullani's Integral
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Theorem
Let $a, b > 0$.
Let $f$ be a function continuously differentiable on the non-negative real numbers.
Suppose that $\ds \map f \infty = \lim_{x \mathop\to \infty} \map f x$ exists, and is finite.
Then:
- $\ds \int_0^\infty \frac {\map f {a x} - \map f {b x} } x \rd x = \paren {\map f \infty - \map f 0} \ln \frac a b$
Proof
\(\ds \int_0^\infty \frac {\map f {a x} - \map f {b x} } x \rd x\) | \(=\) | \(\ds \int_0^\infty \intlimits {\frac {\map f {x t} } x} {t = b} a \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^\infty \int_b^a \map {f'} {x t} \rd t \rd x\) | Fundamental Theorem of Calculus | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_b^a \int_0^\infty \map {f'} {x t} \rd x \rd t\) | Fubini's Theorem | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_b^a \intlimits {\frac {\map f {x t} } t} {x = 0} \infty \rd t\) | Fundamental Theorem of Calculus | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_b^a \frac {\map f \infty - \map f 0} t \rd t\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\map f \infty - \map f 0} \paren {\ln a -\ln b}\) | Primitive of Reciprocal, Fundamental Theorem of Calculus | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\map f \infty - \map f 0} \ln \frac a b\) | Difference of Logarithms |
$\blacksquare$
Source of Name
This entry was named for Giuliano Frullani.
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 15$: Miscellaneous Definite Integrals: $15.118$
- Weisstein, Eric W. "Frullani's Integral." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FrullanisIntegral.html