What is the integral of (x+27)/(x^2-x-30) ?

The integral of (x+27)/(x^2-x-30) is -2ln|x+5|+3ln|x-6|+C

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integral of (x+27)/(x^2-x-30)

\int \frac{x + 27}{x ^{2} - x - 30} d x

- 2 ln ∣ x + 5 ∣ + 3 ln ∣ x - 6 ∣ + C

Solution steps

\int \frac{x + 27}{x ^{2} - x - 30} d x

Take the partial fraction of

\frac{x + 27}{x ^{2} - x - 30} : - \frac{2}{x + 5} + \frac{3}{x - 6}

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

\int \frac{2}{x + 5} d x = 2 ln ∣ x + 5 ∣

\int \frac{3}{x - 6} d x = 3 ln ∣ x - 6 ∣

= - 2 ln ∣ x + 5 ∣ + 3 ln ∣ x - 6 ∣

Add a constant to the solution

= - 2 ln ∣ x + 5 ∣ + 3 ln ∣ x - 6 ∣ + C

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What is the integral of (x+27)/(x^2-x-30) ?
The integral of (x+27)/(x^2-x-30) is -2ln|x+5|+3ln|x-6|+C