What is the integral of ln(x^2-4x+6) ?

The integral of ln(x^2-4x+6) is xln(x^2-4x+6)-2ln(x^2-4x+6)+2sqrt(2)arctan(1/(sqrt(2))(x-2))-2x+4+C

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integral of ln(x^2-4x+6)

\int ln (x^{2} - 4 x + 6) d x

x ln (x^{2} - 4 x + 6) - 2 ln (x^{2} - 4 x + 6) + 22 arctan (\frac{1}{2} (x - 2)) - 2 x + 4 + C

Solution steps

\int ln (x^{2} - 4 x + 6) d x

Complete the square

x^{2} - 4 x + 6 : (x - 2)^{2} + 2

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

Apply u-substitution

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

= \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

= 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 - 2

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

Simplify

(x - 2) ln ((x - 2)^{2} + 2) - (- 22 arctan (\frac{x - 2}{2}) + 2 (x - 2)) : x ln (x^{2} - 4 x + 6) - 2 ln (x^{2} - 4 x + 6) + 22 arctan (\frac{1}{2} (x - 2)) - 2 x + 4

= x ln (x^{2} - 4 x + 6) - 2 ln (x^{2} - 4 x + 6) + 22 arctan (\frac{1}{2} (x - 2)) - 2 x + 4

Add a constant to the solution

= x ln (x^{2} - 4 x + 6) - 2 ln (x^{2} - 4 x + 6) + 22 arctan (\frac{1}{2} (x - 2)) - 2 x + 4 + C

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What is the integral of ln(x^2-4x+6) ?
The integral of ln(x^2-4x+6) is xln(x^2-4x+6)-2ln(x^2-4x+6)+2sqrt(2)arctan(1/(sqrt(2))(x-2))-2x+4+C