If-f-3-3-f-1-2-1-3-f-x-f-x-dx-

Question Number 38405 by malwaan last updated on 25/Jun/18

$$\mathrm{If}\:\mathrm{f}\left(\mathrm{3}\right)=\mathrm{3};\mathrm{f}\left(\mathrm{1}\right)=\mathrm{2}\: \\ $$$$\Rightarrow\int_{\mathrm{1}} ^{\mathrm{3}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{f}^{'} \left(\mathrm{x}\right)\mathrm{dx}=….. \\ $$

Commented by tanmay.chaudhury50@gmail.com last updated on 25/Jun/18

$${pls}\:{clarify}\:\left[\left({x}\right)\right]\:{is}\:{it}\:{floor}/{greatest}\:{integer} \\ $$$${function} \\ $$

Commented by malwaan last updated on 25/Jun/18

$$\mathrm{I}\:\mathrm{am}\:\mathrm{sorry} \\ $$$$\mathrm{It}\:\mathrm{is}\:\mathrm{f}\left(\mathrm{x}\right) \\ $$

Answered by tanmay.chaudhury50@gmail.com last updated on 25/Jun/18

$$\int_{\mathrm{1}} ^{\mathrm{3}} {f}\left({x}\right)×\frac{{df}\left({x}\right)}{{dx}}{dx} \\ $$$$\int_{\mathrm{1}} ^{\mathrm{3}} {f}\left({x}\right){df}\left({x}\right) \\ $$$$\mid\frac{\left\{{f}\left({x}\right)\right\}^{\mathrm{2}} }{\mathrm{2}}\mid_{\mathrm{1}} ^{\mathrm{3}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left\{{f}\left(\mathrm{3}\right)×{f}\left(\mathrm{3}\right)−{f}\left(\mathrm{1}\right)×{f}\left(\mathrm{1}\right)\right\} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{3}^{\mathrm{2}} −\mathrm{2}^{\mathrm{2}} \right)=\frac{\mathrm{5}}{\mathrm{2}} \\ $$

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