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Question Number 125500 by mathmax by abdo last updated on 11/Dec/20

$$\mathrm{find}\mathrm{relation}\mathrm{between}\int \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}\mathrm{and}\int {\mathrm{f}}^{-1}\left(\mathrm{x}\right)\mathrm{dx}$$

Answered by mathmax by abdo last updated on 11/Dec/20

$$\mathrm{let}[\mathrm{a},\mathrm{b}]\subset \stackrel{-}{\mathrm{R}}\mathrm{we}\mathrm{considere}\mathrm{I}={\int }_{\mathrm{a}}^{\mathrm{b}}{\mathrm{f}}^{-1}\left(\mathrm{x}\right)\mathrm{dx}\mathrm{we}\mathrm{do}\mathrm{the}\mathrm{changement}$$$${\mathrm{f}}^{-1}\left(\mathrm{x}\right)=\mathrm{t}\Rightarrow \mathrm{x}=\mathrm{f}\left(\mathrm{t}\right)\Rightarrow \mathrm{I}={\int }_{{\mathrm{f}}^{-1}\left(\mathrm{a}\right)}^{{\mathrm{f}}^{-1}\left(\mathrm{b}\right)}\mathrm{t}{\mathrm{f}}^{{}^{\prime }}\left(\mathrm{t}\right)\mathrm{dt}{=}_{\mathrm{by}\mathrm{parts}}$$$${\left[\mathrm{tf}\left(\mathrm{t}\right)\right]}_{{\mathrm{f}}^{-1}\left(\mathrm{a}\right)}^{{\mathrm{f}}^{-1}\left(\mathrm{b}\right)}-{\int }_{{\mathrm{f}}^{-1}\left(\mathrm{a}\right)}^{{\mathrm{f}}^{-1}\left(\mathrm{b}\right)}\mathrm{f}\left(\mathrm{t}\right)\mathrm{dt}={\mathrm{bf}}^{-1}\left(\mathrm{b}\right)-{\mathrm{af}}^{-1}\left(\mathrm{a}\right)-{\int }_{{\mathrm{f}}^{-1}\left(\mathrm{a}\right)}^{{\mathrm{f}}^{-1}\left(\mathrm{b}\right)}\mathrm{f}\left(\mathrm{t}\right)\mathrm{dt}\Rightarrow$$$${\int }_{\mathrm{a}}^{\mathrm{b}}{\mathrm{f}}^{-1}\left(\mathrm{x}\right)={\mathrm{bf}}^{-1}\left(\mathrm{b}\right)-{\mathrm{af}}^{-1}\left(\mathrm{a}\right)-{\int }_{{\mathrm{f}}^{-1}\left(\mathrm{a}\right)}^{{\mathrm{f}}^{-1}\left(\mathrm{b}\right)}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$$

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