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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:[clarification needed]
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.[1]
References
G. Boros, V. Moll, Irresistible Integrals (2004), pp. 98