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

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

- ln (∣ x ∣ ∣ 1 - ln (x) ∣) + C

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - ln ∣ u ∣

Substitute back

u = x (1 - ln (x))

= - ln ∣ x (1 - ln (x)) ∣

Simplify

- ln ∣ x (1 - ln (x)) ∣ : - ln (∣ x ∣ ∣ 1 - ln (x) ∣)

= - ln (∣ x ∣ ∣ 1 - ln (x) ∣)

Add a constant to the solution

= - ln (∣ x ∣ ∣ 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 -ln(|x||1-ln(x)|)+C