What is the integral of (ln(x+1))/((x+2)^2) ?

The integral of (ln(x+1))/((x+2)^2) is -(ln(x+1))/(x+2)+ln|x+1|-ln|x+2|+C

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integral of (ln(x+1))/((x+2)^2)

\int \frac{ln ( x + 1 )}{( x + 2 ) ^{2}} d x

- \frac{ln ( x + 1 )}{x + 2} + ln ∣ x + 1 ∣ - ln ∣ x + 2 ∣ + C

Solution steps

\int \frac{ln ( x + 1 )}{( x + 2 ) ^{2}} d x

Apply Integration By Parts

: - \frac{ln ( x + 1 )}{x + 2} - \int - \frac{1}{( x + 1 ) ( x + 2 )} d x

= - \frac{ln ( x + 1 )}{x + 2} - \int - \frac{1}{( x + 1 ) ( x + 2 )} d x

\int - \frac{1}{( x + 1 ) ( x + 2 )} d x = - ln ∣ x + 1 ∣ + ln ∣ x + 2 ∣

= - \frac{ln ( x + 1 )}{x + 2} - (- ln ∣ x + 1 ∣ + ln ∣ x + 2 ∣)

Simplify

= - \frac{ln ( x + 1 )}{x + 2} + ln ∣ x + 1 ∣ - ln ∣ x + 2 ∣

Add a constant to the solution

= - \frac{ln ( x + 1 )}{x + 2} + ln ∣ x + 1 ∣ - ln ∣ x + 2 ∣ + C

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What is the integral of (ln(x+1))/((x+2)^2) ?
The integral of (ln(x+1))/((x+2)^2) is -(ln(x+1))/(x+2)+ln|x+1|-ln|x+2|+C