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Questions  >>  JEEMAIN and NEET  >>  Mathematics  >>  Class12  >>  Integral Calculus
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Q)

Evaluate $ \Large \int $$ \large \frac{\log x }{(1+\log x)^2}$$dx$

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A)
Substitute $u = \log x$ and $du = \large\frac{1}{x}$$ dx$
$\Rightarrow I = \Large \int $$ \large\frac{ue^u}{(1+u^2)}$$ du$
Integrating by parts, we get $I = \Large\int $$e^u\;du - \large\frac{ ue^u}{1+u}$
Solving, we get $I = e^u - \large\frac{ue^u}{1+u}$ ($\large\int $$e^u = e^u$)
Substituting back $u = \log x \rightarrow I = \large\frac{x}{1+\log x} $$ + c$
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