The value of 1 x+x dx is: - Vidyadip

MCQ

The value of $\int\frac{1}{\text{x}+\text{x}\log\text{x}}\text{ dx}$ is:

A
$1+\log\text{x}$
B
$\text{x}+\log\text{x}$
C
$\text{x}\log\text{x}(1+\log\text{x})$
✓
$\log(1+\log\text{x})$

✓

Answer

Correct option: D.

$\log(1+\log\text{x})$

$\text{I}=\int\frac{1}{\text{x}+\text{x}\log\text{x}}\text{ dx}$

$\text{I}=\int\frac{\text{dx}}{\text{x}+(1+\log\text{x})}$

Put $1+\log\text{x}=\text{t}$

$\frac{1}{\text{x}}\text{dx}=\text{dt}$

$\text{I}=\int\frac{1}{\text{t}}\text{ dt}$

$\text{I}=\log|\text{t}|+\text{C}$

$\text{I}=\log(1+\log\text{x})+\text{C}$

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