What is the integral of (3x-1)/(x^2-x-6) ?

The integral of (3x-1)/(x^2-x-6) is 7/5 ln|x+2|+8/5 ln|x-3|+C

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integral of (3x-1)/(x^2-x-6)

\int \frac{3 x - 1}{x ^{2} - x - 6} d x

\frac{7}{5} ln ∣ x + 2 ∣ + \frac{8}{5} ln ∣ x - 3 ∣ + C

Solution steps

\int \frac{3 x - 1}{x ^{2} - x - 6} d x

Take the partial fraction of

\frac{3 x - 1}{x ^{2} - x - 6} : \frac{7}{5 ( x + 2 )} + \frac{8}{5 ( x - 3 )}

= \int \frac{7}{5 ( x + 2 )} + \frac{8}{5 ( x - 3 )} 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{7}{5 ( x + 2 )} d x + \int \frac{8}{5 ( x - 3 )} d x

\int \frac{7}{5 ( x + 2 )} d x = \frac{7}{5} ln ∣ x + 2 ∣

\int \frac{8}{5 ( x - 3 )} d x = \frac{8}{5} ln ∣ x - 3 ∣

= \frac{7}{5} ln ∣ x + 2 ∣ + \frac{8}{5} ln ∣ x - 3 ∣

Add a constant to the solution

= \frac{7}{5} ln ∣ x + 2 ∣ + \frac{8}{5} ln ∣ x - 3 ∣ + C

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What is the integral of (3x-1)/(x^2-x-6) ?
The integral of (3x-1)/(x^2-x-6) is 7/5 ln|x+2|+8/5 ln|x-3|+C