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

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

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

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

- x ln (1 - x) - 1 + x + ln ∣ 1 - x ∣ + C

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{x}{1 - x} d x = 1 - x - ln ∣ 1 - x ∣

= - x ln (1 - x) - (1 - x - ln ∣ 1 - x ∣)

Simplify

= - x ln (1 - x) - 1 + x + ln ∣ 1 - x ∣

Add a constant to the solution

= - x ln (1 - x) - 1 + x + ln ∣ 1 - x ∣ + C

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