Question

Integrate: ${\int }_{1/e}^e\left|x\right|dx=$

Answer 1

Consider the given integral.

$I={\int }_e^{1/e}\frac{\ln x}{x}dx$

Let $t=\ln x$

$dt=\ln xdx$

Therefore,

$I={\int }_1^{-1}tdt$

$I={\left[\frac{t^2}{2}\right]}_1^{-1}$

$I=[\frac{{(-1)}^2}{2}-\frac{1^2}{2}]$

$I=0$

Hence, this is the answer.