Evaluate ∫log⁡$$x(1+log$$⁡$$x)2dx$$

Correct option is 3x(log⁡x+1)+c

Questions

$Evaluate \int \frac{\log x}{(1 + \log x)^{2}} d x$

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-x(log⁡x+1)+c

-x(log⁡x+1)2+c

x(log⁡x+1)+c

None of these

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detailed solution

Correct option is C

I=∫log⁡x(1+log⁡x)2dx Let log⁡x=t . Then x=et or dx=etdt . Thus, I=∫tet(t+1)2dt=∫(t+1-1)et(t+1)2dt (add and subtract et in the numerator) =∫et1(t+1)−1(t+1)2dt=ett+1+c=x(log⁡x+1)+c using the formula (∫exf(x)+f′(x)dx=exf(x)+C)

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Evaluate ∫log⁡x(1+log⁡x)2dx

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$Evaluate \int \frac{\log x}{(1 + \log x)^{2}} d x$