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

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

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

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

- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣ + C

Solution steps

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

Take the partial fraction of

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

= \int - \frac{1}{5 ( x + 2 )} + \frac{1}{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{1}{5 ( x + 2 )} d x + \int \frac{1}{5 ( x - 3 )} d x

\int \frac{1}{5 ( x + 2 )} d x = \frac{1}{5} ln ∣ x + 2 ∣

\int \frac{1}{5 ( x - 3 )} d x = \frac{1}{5} ln ∣ x - 3 ∣

= - \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣

Add a constant to the solution

= - \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣ + C

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