What is the integral of (5x+8)/(x^2-x-2) ?

The integral of (5x+8)/(x^2-x-2) is -ln|x+1|+6ln|x-2|+C

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integral of (5x+8)/(x^2-x-2)

\int \frac{5 x + 8}{x ^{2} - x - 2} d x

- ln ∣ x + 1 ∣ + 6 ln ∣ x - 2 ∣ + C

Solution steps

\int \frac{5 x + 8}{x ^{2} - x - 2} d x

Take the partial fraction of

\frac{5 x + 8}{x ^{2} - x - 2} : - \frac{1}{x + 1} + \frac{6}{x - 2}

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

\int \frac{1}{x + 1} d x = ln ∣ x + 1 ∣

\int \frac{6}{x - 2} d x = 6 ln ∣ x - 2 ∣

= - ln ∣ x + 1 ∣ + 6 ln ∣ x - 2 ∣

Add a constant to the solution

= - ln ∣ x + 1 ∣ + 6 ln ∣ x - 2 ∣ + C

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What is the integral of (5x+8)/(x^2-x-2) ?
The integral of (5x+8)/(x^2-x-2) is -ln|x+1|+6ln|x-2|+C