What is the integral of ((x-5)/(x^2-x-6)) ?

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

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integral of ((x-5)/(x^2-x-6))

\int (\frac{x - 5}{x ^{2} - x - 6}) d x

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

Solution steps

\int \frac{x - 5}{x ^{2} - x - 6} d x

Take the partial fraction of

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

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

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

\int \frac{2}{5 ( x - 3 )} d x = \frac{2}{5} ln ∣ x - 3 ∣

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

Add a constant to the solution

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

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\int \frac{x}{5 - x} d x

\int \frac{5}{36 + ( x - 1 ) ^{2}} d x

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