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Question Number 201464 by hardmath last updated on 06/Dec/23

$$\mathrm{if}\mathrm{f}\left(2\right)=3\mathrm{and}\mathrm{f}\left(4\right)=5$$$$\mathrm{find}{\int }_2^4\mathrm{f}\left(\mathrm{x}\right)\cdot \mathrm{f}{}^{{}^{\prime }}\left(\mathrm{x}\right)\mathrm{dx}=?$$

Answered by mahdipoor last updated on 06/Dec/23

$$={\left[\frac{{\left(f\left(x\right)\right)}^2}{2}\right]}_2^4=\frac{25-9}{2}=8$$

Commented by hardmath last updated on 06/Dec/23

$$\mathrm{thank}\mathrm{you},\mathrm{but}\mathrm{I}{\mathrm{didn}}^{\prime }\mathrm{t}\mathrm{understand}$$

Answered by mr W last updated on 07/Dec/23

$$letu=f\left(x\right)$$$$du=f^{\prime }\left(x\right)dx$$$$atx=2:u=f\left(2\right)=3$$$$atx=4:u=f4)=5$$$${\int }_2^{4}f\left(x\right)f^{\prime }\left(x\right)dx$$$$={\int }_{f\left(2\right)}^{f\left(4\right)}udu$$$$={\left[\frac{u^2}{2}\right]}_{f\left(2\right)}^{f\left(4\right)}$$$$={\left[\frac{u^2}{2}\right]}_3^5$$$$=\frac{5^2-3^2}{2}$$$$=8$$

Commented by Calculusboy last updated on 06/Dec/23

$$\mathit{n}\mathit{i}\mathit{c}\mathit{e}\mathit{s}\mathit{o}\mathit{l}\mathit{u}\mathit{t}\mathit{i}\mathit{o}\mathit{n}$$

Commented by hardmath last updated on 08/Dec/23

$$\mathrm{cool}\mathrm{dear}\mathrm{professor}\mathrm{thank}\mathrm{you}$$

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