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

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

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

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

ln (x^{2} + 2 x + 2) (x + 1) - 2 (- arctan (x + 1) + x + 1) + C

Solution steps

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

Complete the square

x^{2} + 2 x + 2 : (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 = 2 (- arctan (u) + u)

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

Substitute back

u = x + 1

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

Expand

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

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

Add a constant to the solution

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

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