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

The integral from 1 to e of 1/(x(1+ln(x))) is ln(2)

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

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

ln (2)

Decimal Notation

0.69314 \dots

Solution steps

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

Apply u-substitution

: \int_{1}^{2} \frac{1}{u} d u

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= [ln ∣ u ∣]_{1}^{2}

Compute the boundaries:

ln (2)

= ln (2)

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What is the integral from 1 to e of 1/(x(1+ln(x))) ?
The integral from 1 to e of 1/(x(1+ln(x))) is ln(2)