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

The integral from 1 to e^2 of 1/(x(1+ln(x))) is ln(3)

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

\int_{1}^{e^{2}} \frac{1}{x ( 1 + ln ( x ) )} d x

ln (3)

Decimal

1.09861 \dots

Solution steps

\int_{1}^{e^{2}} \frac{1}{x ( 1 + ln ( x ) )} d x

Apply u-substitution

= \int_{1}^{3} \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= [ln ∣ u ∣]_{1}^{3}

Compute the boundaries:

ln (3)

= ln (3)

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What is the integral from 1 to e^2 of 1/(x(1+ln(x))) ?
The integral from 1 to e^2 of 1/(x(1+ln(x))) is ln(3)