# Frullani integral

Frullani integrals are definite integrals of the form

${\displaystyle \int \limits _^{\infty }{\frac }\,{\rm }x}$
where ${\displaystyle }$ is a function over ${\displaystyle }$, and the limit of ${\displaystyle }$ exists at ${\displaystyle {\infty }}$

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

${\displaystyle {\int \limits _^{\infty }{\frac }\,{\rm }x=(f(0)-f(\infty ))\ln {\frac }}}$ .

This can be proved using the method of differentiation under the integral sign when the integral exists and ${\displaystyle f'(x)}$ is continuous.

## References

• G. Boros, V. 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.