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

If f(3)=3;f(1)=2 ⇒∫_1 ^3 f(x)f^′ (x)dx=.....

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

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

pls clarify [(x)] is it floor/greatest integer function

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

Commented by malwaan last updated on 25/Jun/18

I am sorry It is f(x)

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

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

∫_1 ^3 f(x)×((df(x))/dx)dx ∫_1 ^3 f(x)df(x) ∣(({f(x)}^2 )/2)∣_1 ^3 =(1/2){f(3)×f(3)−f(1)×f(1)} (1/2)(3^2 −2^2 )=(5/2)

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

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