What is the integral of (x-8)/(x^2-x-6) ?

The integral of (x-8)/(x^2-x-6) is 2ln|x+2|-ln|x-3|+C

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integral of (x-8)/(x^2-x-6)

\int \frac{x - 8}{x ^{2} - x - 6} d x

2 ln ∣ x + 2 ∣ - ln ∣ x - 3 ∣ + C

Solution steps

\int \frac{x - 8}{x ^{2} - x - 6} d x

Take the partial fraction of

\frac{x - 8}{x ^{2} - x - 6} : \frac{2}{x + 2} - \frac{1}{x - 3}

= \int \frac{2}{x + 2} - \frac{1}{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 \frac{2}{x + 2} d x - \int \frac{1}{x - 3} d x

\int \frac{2}{x + 2} d x = 2 ln ∣ x + 2 ∣

\int \frac{1}{x - 3} d x = ln ∣ x - 3 ∣

= 2 ln ∣ x + 2 ∣ - ln ∣ x - 3 ∣

Add a constant to the solution

= 2 ln ∣ x + 2 ∣ - ln ∣ x - 3 ∣ + C

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What is the integral of (x-8)/(x^2-x-6) ?
The integral of (x-8)/(x^2-x-6) is 2ln|x+2|-ln|x-3|+C