What is the integral of ln|1+x| ?

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

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integral of ln|1+x|

\int ln ∣ 1 + x ∣ d x

x ln ∣ 1 + x ∣ - 1 - x + ln ∣ 1 + x ∣ + C

Solution steps

\int ln ∣ 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+x| ?
The integral of ln|1+x| is xln|1+x|-1-x+ln|1+x|+C