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
:$\int _^\,x$
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:

:$\int _^\,x=\Big(f(\infty)-f(0)\Big)\ln .$

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 $f'(xt) = \frac \left(\frac\right)$:

:\begin
\frac &= \left frac\right^ \, \\
& = \int_b^a f'(xt) \, dt \\
\end

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

:\begin
\int_0^\infty \frac \,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\right^\, dt \\
& = \int_b^a \frac\, dt \\
& = \Big(f(\infty)-f(0)\Big)\Big(\ln(a)-\ln(b)\Big) \\
& = \Big(f(\infty)-f(0)\Big)\ln\Big(\frac\Big) \\
\end
Note that the integral in the second line above has been taken over the interval ,a/math>, not ,b/math>.

Applications

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

:$\Big) = \ln(b)$

The formula can also be generalized in several different ways.

References

* G. Boros, Victor Hugo Moll, Irresistible Integrals (2004), pp. 98
* Juan Arias-de-Reyna
On the Theorem of Frullani
(PDF; 884 kB), Proc. A.M.S. 109 (1990), 165-175.
* ProofWiki
proof of Frullani's integral

{{Lists of integrals

Integrals