What is the integral of (2x^2-x-20)/(x^2+x-6) ?

The integral of (2x^2-x-20)/(x^2+x-6) is 2x-14/5 ln|x-2|-1/5 ln|x+3|+C

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integral of (2x^2-x-20)/(x^2+x-6)

\int \frac{2 x ^{2} - x - 20}{x ^{2} + x - 6} d x

2 x - \frac{14}{5} ln ∣ x - 2 ∣ - \frac{1}{5} ln ∣ x + 3 ∣ + C

Solution steps

\int \frac{2 x ^{2} - x - 20}{x ^{2} + x - 6} d x

Take the partial fraction of

\frac{2 x ^{2} - x - 20}{x ^{2} + x - 6} : 2 - \frac{14}{5 ( x - 2 )} - \frac{1}{5 ( x + 3 )}

= \int 2 - \frac{14}{5 ( x - 2 )} - \frac{1}{5 ( x + 3 )} 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 2 d x - \int \frac{14}{5 ( x - 2 )} d x - \int \frac{1}{5 ( x + 3 )} d x

\int 2 d x = 2 x

\int \frac{14}{5 ( x - 2 )} d x = \frac{14}{5} ln ∣ x - 2 ∣

\int \frac{1}{5 ( x + 3 )} d x = \frac{1}{5} ln ∣ x + 3 ∣

= 2 x - \frac{14}{5} ln ∣ x - 2 ∣ - \frac{1}{5} ln ∣ x + 3 ∣

Add a constant to the solution

= 2 x - \frac{14}{5} ln ∣ x - 2 ∣ - \frac{1}{5} ln ∣ x + 3 ∣ + C

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What is the integral of (2x^2-x-20)/(x^2+x-6) ?
The integral of (2x^2-x-20)/(x^2+x-6) is 2x-14/5 ln|x-2|-1/5 ln|x+3|+C