How to calculate this integral?

$$\int_{\frac{1}{e}}^{e}|\ln x| dx$$

It is an obvious integral if the function is under the integral without modulus. The whole catch is in the module.


1 Answer 1


You can use that

$$ \int_{1/e}^{e}\left|\ln\left(x\right)\right|\text{d}x=\int_{1/e}^{1}\left|\ln\left(x\right)\right|\text{d}x+\int_{1}^{e}\left|\ln\left(x\right)\right|\text{d}x $$ Can you now discuss the value of $\left|\ln\left(x\right)\right|$ in each interval ?

  • $\begingroup$ I think the upper limit of the second integral should be $e$, not $1/e$, right? $\endgroup$
    – an4s
    Mar 26, 2021 at 4:26
  • $\begingroup$ @an4s You are fully right $\endgroup$
    – Atmos
    Mar 26, 2021 at 9:54

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