Mathematics

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


ANSWER

0


SOLUTION

Consider the given integral.

$$I=\int_{e}^{1/e}{\dfrac{\ln x}{x}}dx$$

 

Let $$t=\ln x$$

$$dt=\ln xdx$$

 

Therefore,

$$ I=\int_{1}^{-1}{tdt} $$

$$ I=\left[ \dfrac{{{t}^{2}}}{2} \right]_{1}^{-1} $$

$$ I=\left[ \dfrac{{{\left( -1 \right)}^{2}}}{2}-\dfrac{{{1}^{2}}}{2} \right] $$

$$ I=0 $$

 

Hence, this is the answer.

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