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

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

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

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

x ln (\frac{x}{1 - x}) + ln ∣ - x + 1 ∣ + C

Solution steps

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

Apply Integration By Parts

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

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

= x ln (\frac{x}{1 - x}) - (- ln ∣ - x + 1 ∣)

Simplify

= x ln (\frac{x}{1 - x}) + ln ∣ - x + 1 ∣

Add a constant to the solution

= x ln (\frac{x}{1 - x}) + ln ∣ - x + 1 ∣ + C

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