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 $\displaystyle \map f \infty = \lim_{x \mathop\to \infty} \map f x$ exists, and is finite.

Then:

$\displaystyle \int_0^\infty \frac {\map f {a x} - \map f {b x} } x \rd x = \paren {\map f \infty - \map f 0} \ln \frac a b$


Proof

\(\displaystyle \int_0^\infty \frac {\map f {a x} - \map f {b x} } x \rd x\) \(=\) \(\displaystyle \int_0^\infty \intlimits {\frac {\map f {x t} } x} {t = b} a \rd x\)
\(\displaystyle \) \(=\) \(\displaystyle \int_0^\infty \int_b^a \map {f'} {x t} \rd t \rd x\) Fundamental Theorem of Calculus
\(\displaystyle \) \(=\) \(\displaystyle \int_b^a \int_0^\infty \map {f'} {x t} \rd x \rd t\) Fubini's Theorem
\(\displaystyle \) \(=\) \(\displaystyle \int_b^a \intlimits {\frac {\map f {x t} } t} {x = 0} \infty \rd t\) Fundamental Theorem of Calculus
\(\displaystyle \) \(=\) \(\displaystyle \int_b^a \frac {\map f \infty - \map f 0} t \rd t\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {\map f \infty - \map f 0} \paren {\ln a -\ln b}\) Primitive of Reciprocal, Fundamental Theorem of Calculus
\(\displaystyle \) \(=\) \(\displaystyle \paren {\map f \infty - \map f 0} \ln \frac a b\) Difference of Logarithms

$\blacksquare$


Source of Name

This entry was named for Giuliano Frullani.


Sources