What is the integral of (5x^2-2)/(x^2-2x-24) ?

The integral of (5x^2-2)/(x^2-2x-24) is 5x-39/5 ln|x+4|+89/5 ln|x-6|+C

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integral of (5x^2-2)/(x^2-2x-24)

\int \frac{5 x ^{2} - 2}{x ^{2} - 2 x - 24} d x

5 x - \frac{39}{5} ln ∣ x + 4 ∣ + \frac{89}{5} ln ∣ x - 6 ∣ + C

Solution steps

\int \frac{5 x ^{2} - 2}{x ^{2} - 2 x - 24} d x

Take the partial fraction of

\frac{5 x ^{2} - 2}{x ^{2} - 2 x - 24} : 5 - \frac{39}{5 ( x + 4 )} + \frac{89}{5 ( x - 6 )}

= \int 5 - \frac{39}{5 ( x + 4 )} + \frac{89}{5 ( 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 5 d x - \int \frac{39}{5 ( x + 4 )} d x + \int \frac{89}{5 ( x - 6 )} d x

\int 5 d x = 5 x

\int \frac{39}{5 ( x + 4 )} d x = \frac{39}{5} ln ∣ x + 4 ∣

\int \frac{89}{5 ( x - 6 )} d x = \frac{89}{5} ln ∣ x - 6 ∣

= 5 x - \frac{39}{5} ln ∣ x + 4 ∣ + \frac{89}{5} ln ∣ x - 6 ∣

Add a constant to the solution

= 5 x - \frac{39}{5} ln ∣ x + 4 ∣ + \frac{89}{5} ln ∣ x - 6 ∣ + C

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What is the integral of (5x^2-2)/(x^2-2x-24) ?
The integral of (5x^2-2)/(x^2-2x-24) is 5x-39/5 ln|x+4|+89/5 ln|x-6|+C