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

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

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integral of ln(x^2+2x)

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

x ln (x^{2} + 2 x) + ln (x^{2} + 2 x) + ln ∣ x + 2 ∣ - ln ∣ x ∣ - 2 x - 2 + C

Solution steps

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

Complete the square

x^{2} + 2 x : (x + 1)^{2} - 1

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

Apply u-substitution

: \int ln (u^{2} - 1) d u

= \int ln (u^{2} - 1) d u

Apply Integration By Parts

: u ln (u^{2} - 1) - \int \frac{2 u ^{2}}{u ^{2} - 1} d u

= u ln (u^{2} - 1) - \int \frac{2 u ^{2}}{u ^{2} - 1} d u

\int \frac{2 u ^{2}}{u ^{2} - 1} d u = - ln ∣ u + 1 ∣ + ln ∣ u - 1 ∣ + 2 u

= u ln (u^{2} - 1) - (- ln ∣ u + 1 ∣ + ln ∣ u - 1 ∣ + 2 u)

Substitute back

u = x + 1

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

Simplify

(x + 1) ln ((x + 1)^{2} - 1) - (- ln ∣ x + 1 + 1 ∣ + ln ∣ x + 1 - 1 ∣ + 2 (x + 1)) : x ln (x^{2} + 2 x) + ln (x^{2} + 2 x) + ln ∣ x + 2 ∣ - ln ∣ x ∣ - 2 x - 2

= x ln (x^{2} + 2 x) + ln (x^{2} + 2 x) + ln ∣ x + 2 ∣ - ln ∣ x ∣ - 2 x - 2

Add a constant to the solution

= x ln (x^{2} + 2 x) + ln (x^{2} + 2 x) + ln ∣ x + 2 ∣ - ln ∣ x ∣ - 2 x - 2 + C

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