Question Number 29454 by prof Abdo imad last updated on 08/Feb/18
![f is a function increasing and C^1 on [a,b] prove ∫_(f(a)) ^(f(b)) f^(−1) (t)dt = ∫_a ^b x f^′ (x)dx](Q29454.png)
$${f}\:{is}\:{a}\:{function}\:{increasing}\:{and}\:{C}^{\mathrm{1}} {on}\:\left[{a},{b}\right]\:{prove} \\ $$$$\:\int_{{f}\left({a}\right)} ^{{f}\left({b}\right)} \:{f}^{−\mathrm{1}} \left({t}\right){dt}\:=\:\int_{{a}} ^{{b}} \:{x}\:{f}^{'} \left({x}\right){dx}\: \\ $$
Commented by prof Abdo imad last updated on 27/Feb/18
$${let}\:{put}\:{f}^{−\mathrm{1}} \left({t}\right)={x}\:\Rightarrow{t}={f}\left({x}\right)\Rightarrow{dt}\:={f}^{'} \left({x}\right){dx} \\ $$$$\int_{{f}\left({a}\right)} ^{{f}\left({b}\right)} \:{f}^{−\mathrm{1}} \left({t}\right){dt}=\:\int_{{a}} ^{{b}\:} \:{x}\:{f}^{'} \left({x}\right){dx}\:. \\ $$