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Question Number 208129 by efronzo1 last updated on 06/Jun/24

$$\underbrace{\pm \cancel{}}$$

Answered by mr W last updated on 06/Jun/24

$$f\left(0\right)=1$$$$f\left(1\right)=379$$$${\int }_1^{379}{f}^{-1}\left(x\right)dx=379\times 1-1\times 0-{\int }_0^{1}f\left(x\right)dx$$$$=379-{[x^{304}+x^{74}+x]}_0^1$$$$=379-3=376$$

Commented by efronzo1 last updated on 06/Jun/24

$$$$$$\mathrm{not}\mathrm{the}\mathrm{answer}\mathrm{is}376$$

Commented by MM42 last updated on 06/Jun/24

$${\int }_a^{b}f\left(x\right)dx\ne {\int }_{f^{-1}\left(a\right)}^{f^{-1}\left(b\right)}{f}^{-1}\left(x\right)dx$$$$forexample$$$$f\left(x\right)=x^{3}f\left(1\right)=1\mathrm{\&}f\left(2\right)=8$$$${{\int }_1^2{x}^{3}dx=\frac{1}{4}{x}^4]}_1^2=\frac{15}{4}$$$${{\int }_1^8{f}^{-1}\left(x\right)dx={\int }_1^8{x}^{\frac{1}{3}}dx=\frac{3}{4}{x}^{\frac{4}{3}}]}_1^8=\frac{3}{4}(16-1)=\frac{45}{4}$$$$\Rightarrow {\int }_1^8{x}^{\frac{1}{3}}dx\ne {\int }_1^2{x}^{3}dx$$$$$$

Commented by mr W last updated on 06/Jun/24

$$ihavecorrected.$$

Commented by mr W last updated on 06/Jun/24

Commented by mr W last updated on 06/Jun/24

$${\int }_c^d{f}^{-1}\left(x\right)dx+{\int }_a^{b}f\left(x\right)dx=bd-ac$$

Commented by MM42 last updated on 06/Jun/24

$$\cancel{⋚}$$

Answered by efronzo1 last updated on 06/Jun/24

$$\mathrm{let}{\mathrm{f}}^{-1}\left(\mathrm{x}\right)=\mathrm{u}\Rightarrow \mathrm{x}=\mathrm{f}\left(\mathrm{u}\right)$$$$\mathrm{dx}=\mathrm{f}{}^{{}^{\prime }}\left(\mathrm{u}\right)\mathrm{du}=\left(304\right)\left(303\right){\mathrm{u}}^{302}+\left(74\right)\left(73\right){\mathrm{u}}^{72}\mathrm{du}$$$$\mathrm{x}=1\Rightarrow \mathrm{f}\left(\mathrm{u}\right)=1;\mathrm{u}=0$$$$\mathrm{x}=379\Rightarrow \mathrm{f}\left(\mathrm{u}\right)=379;\mathrm{u}=1$$$$\mathrm{I}=\underset{1}{\stackrel{379}{\int }}{\mathrm{f}}^{-1}\left(\mathrm{x}\right)\mathrm{dx}=\underset{0}{\stackrel{1}{\int }}\mathrm{u}(\left(304\right)\left(303\right){\mathrm{u}}^{302}+\left(74\right)\left(73\right){\mathrm{u}}^{72})\mathrm{du}$$$$=\underset{0}{\stackrel{1}{\int }}\left(304\right)\left(303\right){\mathrm{u}}^{303}+\left(74\right)\left(73\right){\mathrm{u}}^{73}\mathrm{du}$$$$={[303{\mathrm{u}}^{304}+73{\mathrm{u}}^{74}]}_0^1$$$$=303+73=376$$$$$$

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