Mark the correct alternative in each of the following:

Question:

Mark the correct alternative in each of the following:

The value of $\int \frac{1}{x+x \log x} d x$ is

A. $1+\log x$

B. $x+\log x$

C. $x \log (1+\log x)$

D. $\log (1+\log x)$

Solution:

$I=\int \frac{1}{x\left(1+\log _{e} x\right)} d \chi$

$\Rightarrow \operatorname{let}\left(1+\log _{e} x\right)=t\left[\frac{d t}{d x}=\frac{1}{x}\right]$

$\Rightarrow \int \frac{1}{t} d t=\log _{e} t$

$\Rightarrow 1=\log (1+\log x)+C$

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