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

= 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

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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