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

The integral of (ln(x))/(x(1+ln(x))) is 1+ln(x)-ln|1+ln(x)|+C

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integral of (ln(x))/(x(1+ln(x)))

\int \frac{ln ( x )}{x ( 1 + ln ( x ) )} d x

1 + ln (x) - ln ∣ 1 + ln (x) ∣ + C

Solution steps

\int \frac{ln ( x )}{x ( 1 + ln ( x ) )} d x

Apply u-substitution

: \int \frac{u}{1 + u} d u

= \int \frac{u}{1 + u} d u

Apply u-substitution

: \int \frac{v - 1}{v} d v

= \int \frac{v - 1}{v} d v

Expand

\frac{v - 1}{v} : 1 - \frac{1}{v}

= \int 1 - \frac{1}{v} d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 1 d v - \int \frac{1}{v} d v

\int 1 d v = v

\int \frac{1}{v} d v = ln ∣ v ∣

= v - ln ∣ v ∣

Substitute back

= 1 + ln (x) - ln ∣ 1 + ln (x) ∣

Add a constant to the solution

= 1 + ln (x) - ln ∣ 1 + ln (x) ∣ + C

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What is the integral of (ln(x))/(x(1+ln(x))) ?
The integral of (ln(x))/(x(1+ln(x))) is 1+ln(x)-ln|1+ln(x)|+C