# Primitive function/Inverse function/Section

## Theorem

Let ${\displaystyle {}f\colon [a,b]\rightarrow [c,d]}$ denote a bijective differentiable function, and let ${\displaystyle {}F}$ denote a primitive function for ${\displaystyle {}f}$. Then

${\displaystyle {}G(y):=yf^{-1}(y)-F{\left(f^{-1}(y)\right)}\,}$

is a primitive function for the inverse function ${\displaystyle {}f^{-1}}$.

### Proof

Differentiating, using fact and fact, yields

{\displaystyle {}{\begin{aligned}{\left(yf^{-1}(y)-F{\left(f^{-1}(y)\right)}\right)}'&=f^{-1}(y)+y{\frac {1}{f'(f^{-1}(y))}}-f{\left(f^{-1}(y)\right)}{\frac {1}{f'{\left(f^{-1}(y)\right)}}}\\&=f^{-1}(y).\end{aligned}}}
${\displaystyle \Box }$

For this statement, there exists also an easy geometric explanation. When ${\displaystyle {}f\colon [a,b]\rightarrow \mathbb {R} _{+}}$ is a strictly increasing continuous function (and therefore induces a bijection between ${\displaystyle {}[a,b]}$ and ${\displaystyle {}[f(a),f(b)]}$), then the following relation between the areas holds:

${\displaystyle {}\int _{a}^{b}f(s)\,ds+\int _{f(a)}^{f(b)}f^{-1}(t)\,dt=bf(b)-af(a)\,,}$

or, equivalently,

${\displaystyle {}\int _{f(a)}^{f(b)}f^{-1}(t)\,dt=bf(b)-af(a)-\int _{a}^{b}f(s)\,ds\,.}$

For the primitive function ${\displaystyle {}G}$ of ${\displaystyle {}f^{-1}}$ with starting point ${\displaystyle {}f(a)}$, we have, if ${\displaystyle {}F}$ denotes a primitive function for ${\displaystyle {}f}$, the relation

{\displaystyle {}{\begin{aligned}G(y)&=\int _{f(a)}^{y}f^{-1}(t)\,dt\\&=\int _{f(a)}^{f(f^{-1}(y))}f^{-1}(t)\,dt\\&=f^{-1}(y)f(f^{-1}(y))-af(a)-\int _{a}^{f^{-1}(y)}f(s)\,ds\\&=yf^{-1}(y)-af(a)-F(f^{-1}(y))+F(a)\\&=yf^{-1}(y)-F(f^{-1}(y))-af(a)+F(a),\end{aligned}}}

where ${\displaystyle {}-af(a)+F(a)}$ is a constant of integration.

## Example

We compute a primitive function for ${\displaystyle {}\arctan x}$, using fact. A primitive function of tangent is

${\displaystyle {}\int _{}^{}\tan t\,dt=-\ln(\cos x)\,.}$

Hence,

${\displaystyle x\cdot \arctan x+\ln(\cos(\arctan x))}$

is a primitive function for ${\displaystyle {}\arctan x}$.