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Question Number 202415 by MathematicalUser2357 last updated on 26/Dec/23

$$\mathrm{The}\mathrm{value}\mathrm{of}\int g^{\prime }\left(x\right)f^{\prime }\left(g\left(x\right)\right)dx\mathrm{is}...$$

Answered by cortano12 last updated on 26/Dec/23

$$\mathrm{let}\mathrm{u}=\mathrm{g}\left(\mathrm{x}\right)\Rightarrow \mathrm{du}={\mathrm{g}}^{\prime }\left(\mathrm{x}\right)\mathrm{dx}$$$$\mathrm{I}=\int {\mathrm{g}}^{\prime }\left(\mathrm{x}\right)\mathrm{f}{}^{\prime }\left(\mathrm{g}\left(\mathrm{x}\right)\right)\mathrm{dx}$$$$=\int \mathrm{f}{}^{\prime }\left(\mathrm{u}\right)\mathrm{du}=\int \frac{\mathrm{df}\left(\mathrm{u}\right)}{\mathrm{du}}.\mathrm{du}$$$$=\int \mathrm{df}\left(\mathrm{u}\right)=\mathrm{f}\left(\mathrm{u}\right)+\mathrm{c}$$$$=\mathrm{f}\left(\mathrm{g}\left(\mathrm{x}\right)\right)+\mathrm{c}$$

Answered by Mathspace last updated on 26/Dec/23

$$=\int {\left(fog\right)}^{{}^{\prime }}\left(x\right)dx=fog\left(x\right)+c$$

Commented by MathematicalUser2357 last updated on 27/Dec/23

$$f\circ g\mathrm{not}\mathrm{fog}$$

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