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Question Number 115267 by bobhans last updated on 24/Sep/20

$$Iff\left(x\right)isadifferentiablefunction$$$$defined\mathrm{\forall }x\in \mathbb{R}suchthat{\left(f\left(x\right)\right)}^3-x+f\left(x\right)=0$$$$then\underset{0}{\stackrel{\sqrt{2}}{\int }}{f}^{-1}\left(x\right)dx=$$

Answered by Olaf last updated on 24/Sep/20

$$f^3\left(t\right)-t+f\left(t\right)=0$$$$\mathrm{Let}t=f^{-1}\left(x\right)$$$${\left(f\mathrm{o}{f}^{-1}\right)}^3\left(x\right)-f^{-1}\left(x\right)+f\mathrm{o}{f}^{-1}\left(x\right)=0$$$$x^3-f^{-1}\left(x\right)+x=0$$$$f^{-1}\left(x\right)=x^3+x$$$${\int }_0^{\sqrt{2}}{f}^{-1}\left(x\right)dx={[\frac{x^4}{4}+\frac{x^2}{2}]}_0^{\sqrt{2}}=1+1=2$$

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