What is the integral of (2x-1)/(x^2-2x-24) ?

The integral of (2x-1)/(x^2-2x-24) is 9/10 ln|x+4|+11/10 ln|x-6|+C

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integral of (2x-1)/(x^2-2x-24)

\int \frac{2 x - 1}{x ^{2} - 2 x - 24} d x

\frac{9}{10} ln ∣ x + 4 ∣ + \frac{11}{10} ln ∣ x - 6 ∣ + C

Solution steps

\int \frac{2 x - 1}{x ^{2} - 2 x - 24} d x

Take the partial fraction of

\frac{2 x - 1}{x ^{2} - 2 x - 24} : \frac{9}{10 ( x + 4 )} + \frac{11}{10 ( x - 6 )}

= \int \frac{9}{10 ( x + 4 )} + \frac{11}{10 ( 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{9}{10 ( x + 4 )} d x + \int \frac{11}{10 ( x - 6 )} d x

\int \frac{9}{10 ( x + 4 )} d x = \frac{9}{10} ln ∣ x + 4 ∣

\int \frac{11}{10 ( x - 6 )} d x = \frac{11}{10} ln ∣ x - 6 ∣

= \frac{9}{10} ln ∣ x + 4 ∣ + \frac{11}{10} ln ∣ x - 6 ∣

Add a constant to the solution

= \frac{9}{10} ln ∣ x + 4 ∣ + \frac{11}{10} ln ∣ x - 6 ∣ + C

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What is the integral of (2x-1)/(x^2-2x-24) ?
The integral of (2x-1)/(x^2-2x-24) is 9/10 ln|x+4|+11/10 ln|x-6|+C