What is the integral of (x^2-9x+11)/(x+6) ?

The integral of (x^2-9x+11)/(x+6) is ((x+6)^2)/2-21(x+6)+101ln|x+6|+C

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integral of (x^2-9x+11)/(x+6)

\int \frac{x ^{2} - 9 x + 11}{x + 6} d x

\frac{( x + 6 ) ^{2}}{2} - 21 (x + 6) + 101 ln ∣ x + 6 ∣ + C

Solution steps

\int \frac{x ^{2} - 9 x + 11}{x + 6} d x

Apply u-substitution

: \int \frac{u ^{2} - 21 u + 101}{u} d u

= \int \frac{u ^{2} - 21 u + 101}{u} d u

Expand

\frac{u ^{2} - 21 u + 101}{u} : u - 21 + \frac{101}{u}

= \int u - 21 + \frac{101}{u} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int u d u - \int 21 d u + \int \frac{101}{u} d u

\int u d u = \frac{u ^{2}}{2}

\int 21 d u = 21 u

\int \frac{101}{u} d u = 101 ln ∣ u ∣

= \frac{u ^{2}}{2} - 21 u + 101 ln ∣ u ∣

Substitute back

u = x + 6

= \frac{( x + 6 ) ^{2}}{2} - 21 (x + 6) + 101 ln ∣ x + 6 ∣

Add a constant to the solution

= \frac{( x + 6 ) ^{2}}{2} - 21 (x + 6) + 101 ln ∣ x + 6 ∣ + C

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What is the integral of (x^2-9x+11)/(x+6) ?
The integral of (x^2-9x+11)/(x+6) is ((x+6)^2)/2-21(x+6)+101ln|x+6|+C