Frullani integral
# Frullani integral

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

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

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

The following formula for their general solution holds under certain conditions:

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

## 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 $f'(xt)={\frac {\partial }{\partial t}}\left({\frac {f(xt)}{x}}\right)$:

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

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

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

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

## Applications

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

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

The formula can also be generalized in several different ways.

