In mathematics, Frullani integrals are a specific type of improper integral named after the Italian mathematician Giuliano Frullani. The integrals are of the form

${\displaystyle \int _{0}^{\infty }{\frac {f(ax)-f(bx)}{x))\,{\rm {d))x}$

where ${\displaystyle f}$ is a function defined for all non-negative real numbers that has a limit at ${\displaystyle \infty }$, which we denote by ${\displaystyle f(\infty )}$.

The following formula for their general solution holds under certain conditions:^{[clarification needed]}

${\displaystyle \int _{0}^{\infty }{\frac {f(ax)-f(bx)}{x))\,{\rm {d))x={\Big (}f(\infty )-f(0){\Big )}\ln {\frac {a}{b)).}$

Proof

A simple proof of the formula can be arrived at by using the Fundamental theorem of calculus to express the integrand as an integral of ${\displaystyle f'(xt)={\frac {\partial }{\partial t))\left({\frac {f(xt)}{x))\right)}$:

${\displaystyle {\begin{aligned}{\frac {f(ax)-f(bx)}{x))&=\left[{\frac {f(xt)}{x))\right]_{t=b}^{t=a}\,\\&=\int _{b}^{a}f'(xt)\,dt\\\end{aligned))}$

and then use Tonelli’s theorem to interchange the two integrals:

${\displaystyle {\begin{aligned}\int _{0}^{\infty }{\frac {f(ax)-f(bx)}{x))\,dx&=\int _{0}^{\infty }\int _{b}^{a}f'(xt)\,dt\,dx\\&=\int _{b}^{a}\int _{0}^{\infty }f'(xt)\,dx\,dt\\&=\int _{b}^{a}\left[{\frac {f(xt)}{t))\right]_{x=0}^{x\to \infty }\,dt\\&=\int _{b}^{a}{\frac {f(\infty )-f(0)}{t))\,dt\\&={\Big (}f(\infty )-f(0){\Big )}{\Big (}\ln(a)-\ln(b){\Big )}\\&={\Big (}f(\infty )-f(0){\Big )}\ln {\Big (}{\frac {a}{b)){\Big )}\\\end{aligned))}$

Note that the integral in the second line above has been taken over the interval ${\displaystyle [b,a]}$, not ${\displaystyle [a,b]}$.

Applications

The formula can be used to derive an integral representation for the natural logarithm ${\displaystyle \ln(x)}$ by letting ${\displaystyle f(x)=e^{-x))$ and ${\displaystyle a=1}$:

${\displaystyle {\int _{0}^{\infty }{\frac {e^{-x}-e^{-bx)){x))\,{\rm {d))x={\Big (}\lim _{n\to \infty }{\frac {1}{e^{n))}-e^{0}{\Big )}\ln {\Big (}{\frac {1}{b))}{\Big )}=\ln(b)}$

The formula can also be generalized in several different ways.^[1]