What is the integral of (x^3-6x+5)/(x^2-x-6) ?

The integral of (x^3-6x+5)/(x^2-x-6) is (x^2)/2+x-9/5 ln|x+2|+14/5 ln|x-3|+C

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integral of (x^3-6x+5)/(x^2-x-6)

\int \frac{x ^{3} - 6 x + 5}{x ^{2} - x - 6} d x

\frac{x ^{2}}{2} + x - \frac{9}{5} ln ∣ x + 2 ∣ + \frac{14}{5} ln ∣ x - 3 ∣ + C

Solution steps

\int \frac{x ^{3} - 6 x + 5}{x ^{2} - x - 6} d x

Long division

\frac{x ^{3} - 6 x + 5}{x ^{2} - x - 6} : x + 1 + \frac{x + 11}{x ^{2} - x - 6}

= \int x + 1 + \frac{x + 11}{x ^{2} - x - 6} d x

Apply the Sum Rule:

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

= \int x d x + \int 1 d x + \int \frac{x + 11}{x ^{2} - x - 6} d x

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

\int 1 d x = x

\int \frac{x + 11}{x ^{2} - x - 6} d x = - \frac{9}{5} ln ∣ x + 2 ∣ + \frac{14}{5} ln ∣ x - 3 ∣

= \frac{x ^{2}}{2} + x - \frac{9}{5} ln ∣ x + 2 ∣ + \frac{14}{5} ln ∣ x - 3 ∣

Add a constant to the solution

= \frac{x ^{2}}{2} + x - \frac{9}{5} ln ∣ x + 2 ∣ + \frac{14}{5} ln ∣ x - 3 ∣ + C

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What is the integral of (x^3-6x+5)/(x^2-x-6) ?
The integral of (x^3-6x+5)/(x^2-x-6) is (x^2)/2+x-9/5 ln|x+2|+14/5 ln|x-3|+C