What is the integral of ln(x^2+6x+10) ?

The integral of ln(x^2+6x+10) is ln(x^2+6x+10)(x+3)-2(-arctan(x+3)+x+3)+C

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integral of ln(x^2+6x+10)

\int ln (x^{2} + 6 x + 10) d x

ln (x^{2} + 6 x + 10) (x + 3) - 2 (- arctan (x + 3) + x + 3) + C

Solution steps

\int ln (x^{2} + 6 x + 10) d x

Complete the square

x^{2} + 6 x + 10 : (x + 3)^{2} + 1

= \int ln ((x + 3)^{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 + 3

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

Expand

(x + 3)^{2} + 1 : x^{2} + 6 x + 10

= ln (x^{2} + 6 x + 10) (x + 3) - 2 (- arctan (x + 3) + x + 3)

Add a constant to the solution

= ln (x^{2} + 6 x + 10) (x + 3) - 2 (- arctan (x + 3) + x + 3) + C

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What is the integral of ln(x^2+6x+10) ?
The integral of ln(x^2+6x+10) is ln(x^2+6x+10)(x+3)-2(-arctan(x+3)+x+3)+C