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
, has finite limit at
: A simple proof of the formula (under stronger assumptions than those stated above, namely
) 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
The formula can be used to derive an integral representation for the natural logarithm
: The formula can also be generalized in several different ways.