What is the integral of ln(x^2+13x+42) ?

The integral of ln(x^2+13x+42) is xln((x+6)(x+7))+6ln((x+6)(x+7))-2x-12+ln|x+7|+C

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integral of ln(x^2+13x+42)

\int ln (x^{2} + 13 x + 42) d x

x ln ((x + 6) (x + 7)) + 6 ln ((x + 6) (x + 7)) - 2 x - 12 + ln ∣ x + 7 ∣ + C

Solution steps

\int ln (x^{2} + 13 x + 42) d x

Factor

x^{2} + 13 x + 42 : (x + 6) (x + 7)

= \int ln ((x + 6) (x + 7)) d x

Apply u-substitution

= \int ln (u (u + 1)) d u

Apply Integration By Parts

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

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

= u ln (u (u + 1)) - (2 u - ln ∣ u + 1 ∣)

Substitute back

u = x + 6

= (x + 6) ln ((x + 6) (x + 6 + 1)) - (2 (x + 6) - ln ∣ x + 6 + 1 ∣)

Simplify

(x + 6) ln ((x + 6) (x + 6 + 1)) - (2 (x + 6) - ln ∣ x + 6 + 1 ∣) : x ln ((x + 6) (x + 7)) + 6 ln ((x + 6) (x + 7)) - 2 x - 12 + ln ∣ x + 7 ∣

= x ln ((x + 6) (x + 7)) + 6 ln ((x + 6) (x + 7)) - 2 x - 12 + ln ∣ x + 7 ∣

Add a constant to the solution

= x ln ((x + 6) (x + 7)) + 6 ln ((x + 6) (x + 7)) - 2 x - 12 + ln ∣ x + 7 ∣ + C

What is the integral of ln(x^2+13x+42) ?
The integral of ln(x^2+13x+42) is xln((x+6)(x+7))+6ln((x+6)(x+7))-2x-12+ln|x+7|+C