What is the integral from 1/e to e of (|ln(x)|)/x ?

The integral from 1/e to e of (|ln(x)|)/x is 1

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integral from 1/e to e of (|ln(x)|)/x

\int_{\frac{1}{e}}^{e} \frac{∣ ln ( x ) ∣}{x} d x

1

Solution steps

\int_{\frac{1}{e}}^{e} \frac{∣ ln ( x ) ∣}{x} d x

Apply u-substitution

: \int_{- 1}^{1} ∣ u ∣ d u

= \int_{- 1}^{1} ∣ u ∣ d u

Eliminate Absolutes

= \int_{- 1}^{0} - u d u + \int_{0}^{1} u d u

\int_{- 1}^{0} - u d u = \frac{1}{2}

\int_{0}^{1} u d u = \frac{1}{2}

= \frac{1}{2} + \frac{1}{2}

Simplify

\frac{1}{2} + \frac{1}{2} : 1

= 1

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What is the integral from 1/e to e of (|ln(x)|)/x ?
The integral from 1/e to e of (|ln(x)|)/x is 1