Integral of inverse functions

integration of inverse functions

Integrals of inverse functions can be computed by mean of a formula that expresses the antiderivative of the inverse $f^{-1}$ of a continuous and invertible function $f$, in term of $f^{-1}$ and the antiderivative of $f$.

Formulation of the theorem

Let $f:[a,b]\to [c,d]\subseteq {\overline {\mathbb {R} }}$ be a continuous and invertible function, with $a\geq -\infty$ and $b\leq +\infty$. Then $f$ and $f^{-1}$ have antiderivatives, and if $F$ is an antiderivative of $f$, then the antiderivative of $f^{-1}$ is $G(x)=xf^{-1}(x)-F\circ f^{-1}(x)+C.$

Remark

If $f$ is assumed to be differentiable, then the proof of the above formula follows immediately by differentiation. Nevertheless, it can be shown that this theorem holds even if $f$ is not differentiable.

Examples

1. Assume that $f(x)=\exp(x)$, hence $f^{-1}(x)=\ln(x)$. The formula gives immediately $\int \ln(x)dx=x\ln(x)-x+C$.

2. Similarly, with $f(x)=\cos(x)$ and $f^{-1}(x)=\arccos$, $\int \arccos(x)dx=x\arccos(x)-\sin(\arccos(x))+C.$

3. With $f(x)=\tan(x)$ and $f^{-1}(x)=\arctan(x)$, $\int \arctan(x)dx=x\arctan(x)+\ln |\cos(\arctan(x)|+C$

History

Apparently, this theorem of integration has been discovered in 1955 by Parker, but with the additional assumption that $f$ is differentiable. It seems that the first proof of the correctness of the general theorem was given by Key in 1994. Two rigorous proofs were also given by Bensimhoun in 2013.

Generalization to analytic functions

The above theorem generalizes in the obvious way to analytic functions: Let $U$ and $V$ be two open sets of $\mathbb {C}$, and assume that $f:U\to V$ is holomorphic, and invertible $U\to V$. Then $f$ and $f^{-1}$ have antiderivatives, and if $F$ is an antiderivative of $f$, then the antiderivative of $f^{-1}$ is $G(z)=zf^{-1}(z)-F\circ f^{-1}(z)+C.$

The proof is immediate by (complex) differentiation.

External links

• example.com