Integral of inverse functions

In mathematics, integrals of inverse functions can be computed by means of a formula that expresses the antiderivatives of the inverse

This formula was published in 1905 by Charles-Ange Laisant.

are continuous, they have antiderivatives by the fundamental theorem of calculus.

is an arbitrary real number.

In his 1905 article, Laisant gave three proofs.

, the theorem can be written: The figure on the right is a proof without words of this formula.

Laisant does not discuss the hypotheses necessary to make this proof rigorous, but this can be proved if

is just assumed to be strictly monotone (but not necessarily continuous, let alone differentiable).

are Riemann integrable and the identity follows from a bijection between lower/upper Darboux sums of

[2][3] The antiderivative version of the theorem then follows from the fundamental theorem of calculus in the case when

Laisant's third proof uses the additional hypothesis that

The right-hand side is calculated using integration by parts to be

by the fundamental theorem of calculus.

is invertible, its derivative would vanish in at most countably many points.

is a composition of differentiable functions on each interval

, chain rule could be applied

By continuity and the fundamental theorem of calculus,

is a constant, is a differentiable extension of

One can now use the fundamental theorem of calculus to compute

is not differentiable:[3][4] it suffices, for example, to use the Stieltjes integral in the previous argument.

On the other hand, even though general monotonic functions are differentiable almost everywhere, the proof of the general formula does not follow, unless

[citation needed] In other words: To this end, it suffices to apply the mean value theorem to

Apparently, this theorem of integration was discovered for the first time in 1905 by Charles-Ange Laisant,[1] who "could hardly believe that this theorem is new", and hoped its use would henceforth spread out among students and teachers.

This result was published independently in 1912 by an Italian engineer, Alberto Caprilli, in an opuscule entitled "Nuove formole d'integrazione".

[5] It was rediscovered in 1955 by Parker,[6] and by a number of mathematicians following him.

The general version of the theorem, free from this additional assumption, was proposed by Michael Spivak in 1965, as an exercise in the Calculus,[2] and a fairly complete proof following the same lines was published by Eric Key in 1994.

[3] This proof relies on the very definition of the Darboux integral, and consists in showing that the upper Darboux sums of the function f are in 1-1 correspondence with the lower Darboux sums of f−1.

In 2013, Michael Bensimhoun, estimating that the general theorem was still insufficiently known, gave two other proofs:[4] The second proof, based on the Stieltjes integral and on its formulae of integration by parts and of homeomorphic change of variables, is the most suitable to establish more complex formulae.

The above theorem generalizes in the obvious way to holomorphic functions: Let

be two open and simply connected sets of