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

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

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integral of ln(1/x+1)

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

Simplify

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

Add a constant to the solution

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

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