Frullani integral


Frullani integrals are a specific type of improper integral named after the Italian mathematician Giuliano Frullani. The integrals are of the form
where is a function over, and the limit of exists at.
The following formula for their general solution holds under certain conditions:

Proof

A simple proof of the formula can be arrived at by expanding the integrand into an integral, and then using Fubini's theorem to interchange the two integrals:
Note that the integral in the second line above has been taken over the interval, not.

Applications

The formula can be used to derive an integral representation for the natural logarithm by letting and :
The formula can also be generalized in several different ways.