What is the integral of ln(x^2+2x+3) ?

The integral of ln(x^2+2x+3) is xln(x^2+2x+3)+ln(x^2+2x+3)+2sqrt(2)arctan(1/(sqrt(2))(x+1))-2x-2+C

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integral of ln(x^2+2x+3)

\int ln (x^{2} + 2 x + 3) d x

x ln (x^{2} + 2 x + 3) + ln (x^{2} + 2 x + 3) + 22 arctan (\frac{1}{2} (x + 1)) - 2 x - 2 + C

Solution steps

\int ln (x^{2} + 2 x + 3) d x

Complete the square

x^{2} + 2 x + 3 : (x + 1)^{2} + 2

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

Apply u-substitution

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

Apply Integration By Parts

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

\int \frac{2 u ^{2}}{u ^{2} + 2} d u = - 22 arctan (\frac{u}{2}) + 2 u

= u ln (u^{2} + 2) - (- 22 arctan (\frac{u}{2}) + 2 u)

Substitute back

u = x + 1

= (x + 1) ln ((x + 1)^{2} + 2) - (- 22 arctan (\frac{x + 1}{2}) + 2 (x + 1))

Simplify

(x + 1) ln ((x + 1)^{2} + 2) - (- 22 arctan (\frac{x + 1}{2}) + 2 (x + 1)) : x ln (x^{2} + 2 x + 3) + ln (x^{2} + 2 x + 3) + 22 arctan (\frac{1}{2} (x + 1)) - 2 x - 2

= x ln (x^{2} + 2 x + 3) + ln (x^{2} + 2 x + 3) + 22 arctan (\frac{1}{2} (x + 1)) - 2 x - 2

Add a constant to the solution

= x ln (x^{2} + 2 x + 3) + ln (x^{2} + 2 x + 3) + 22 arctan (\frac{1}{2} (x + 1)) - 2 x - 2 + C

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What is the integral of ln(x^2+2x+3) ?
The integral of ln(x^2+2x+3) is xln(x^2+2x+3)+ln(x^2+2x+3)+2sqrt(2)arctan(1/(sqrt(2))(x+1))-2x-2+C