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
where is a function defined for all non-negative real numbers that has a limit at, which we denote by.
The following formula for their general solution holds if is continuous on, has finite limit at, and :
If does not exist, but exists for some, then

Proof for continuously differentiable functions

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 :
and then use Tonelli’s theorem to interchange the two integrals:
Note that the integral in the second line above has been taken over the interval, not.

Ramanujan's generalization

Ramanujan, using his master theorem, gave the following generalization.
Let be functions continuous on.Let and be given as above, and assume that and are continuous functions on. Also assume that and. Then, if,

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.