What is the integral of (3x+23)/(x^2-8x-9) ?

The integral of (3x+23)/(x^2-8x-9) is -2ln|x+1|+5ln|x-9|+C

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integral of (3x+23)/(x^2-8x-9)

\int \frac{3 x + 23}{x ^{2} - 8 x - 9} d x

- 2 ln ∣ x + 1 ∣ + 5 ln ∣ x - 9 ∣ + C

Solution steps

\int \frac{3 x + 23}{x ^{2} - 8 x - 9} d x

Take the partial fraction of

\frac{3 x + 23}{x ^{2} - 8 x - 9} : - \frac{2}{x + 1} + \frac{5}{x - 9}

= \int - \frac{2}{x + 1} + \frac{5}{x - 9} 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 + 1} d x + \int \frac{5}{x - 9} d x

\int \frac{2}{x + 1} d x = 2 ln ∣ x + 1 ∣

\int \frac{5}{x - 9} d x = 5 ln ∣ x - 9 ∣

= - 2 ln ∣ x + 1 ∣ + 5 ln ∣ x - 9 ∣

Add a constant to the solution

= - 2 ln ∣ x + 1 ∣ + 5 ln ∣ x - 9 ∣ + C

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What is the integral of (3x+23)/(x^2-8x-9) ?
The integral of (3x+23)/(x^2-8x-9) is -2ln|x+1|+5ln|x-9|+C