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f-x-ln-x-x-1-e-f-x-x-dx-1-e-ln-f-x-x-dx-




Question Number 933 by 123456 last updated on 29/Apr/15
f(x)=((ln x)/x)  ∫_1 ^e  ((f(x))/x)dx=?   ∫_1 ^e ((ln f(x))/x)dx=?
$${f}\left({x}\right)=\frac{\mathrm{ln}\:{x}}{{x}} \\ $$$$\underset{\mathrm{1}} {\overset{{e}} {\int}}\:\frac{{f}\left({x}\right)}{{x}}{dx}=? \\ $$$$\:\underset{\mathrm{1}} {\overset{{e}} {\int}}\frac{\mathrm{ln}\:{f}\left({x}\right)}{{x}}{dx}=? \\ $$
Commented by prakash jain last updated on 30/Apr/15
∫ ((ln x)/x^2 ) dx=ln x∙((−1)/x)+∫(1/x^2 ) dx  =−((ln x)/x)−(1/x)  limits from 1 to e,  ((−1)/e)−(1/e)+1=1−(2/e)
$$\int\:\frac{\mathrm{ln}\:{x}}{{x}^{\mathrm{2}} }\:{dx}=\mathrm{ln}\:{x}\centerdot\frac{−\mathrm{1}}{{x}}+\int\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\:{dx} \\ $$$$=−\frac{\mathrm{ln}\:{x}}{{x}}−\frac{\mathrm{1}}{{x}} \\ $$$${limits}\:\mathrm{from}\:\mathrm{1}\:\mathrm{to}\:{e},\:\:\frac{−\mathrm{1}}{{e}}−\frac{\mathrm{1}}{{e}}+\mathrm{1}=\mathrm{1}−\frac{\mathrm{2}}{{e}} \\ $$

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