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

The integral of (x-2)/(x(x^2-x-6)) is -2/5 ln|x+2|+1/15 ln|x-3|+1/3 ln|x|+C

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

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

- \frac{2}{5} ln ∣ x + 2 ∣ + \frac{1}{15} ln ∣ x - 3 ∣ + \frac{1}{3} ln ∣ x ∣ + C

Solution steps

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

Expand

\frac{x - 2}{x ( x ^{2} - x - 6 )} : \frac{x}{x ( x ^{2} - x - 6 )} - \frac{2}{x ( x ^{2} - x - 6 )}

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

\int \frac{x}{x ( x ^{2} - x - 6 )} d x = - \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣

\int \frac{2}{x ( x ^{2} - x - 6 )} d x = 2 (- \frac{1}{6} ln ∣ x ∣ + \frac{1}{10} ln ∣ x + 2 ∣ + \frac{1}{15} ln ∣ x - 3 ∣)

= - \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣ - 2 (- \frac{1}{6} ln ∣ x ∣ + \frac{1}{10} ln ∣ x + 2 ∣ + \frac{1}{15} ln ∣ x - 3 ∣)

Simplify

- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣ - 2 (- \frac{1}{6} ln ∣ x ∣ + \frac{1}{10} ln ∣ x + 2 ∣ + \frac{1}{15} ln ∣ x - 3 ∣) : - \frac{2}{5} ln ∣ x + 2 ∣ + \frac{1}{15} ln ∣ x - 3 ∣ + \frac{1}{3} ln ∣ x ∣

= - \frac{2}{5} ln ∣ x + 2 ∣ + \frac{1}{15} ln ∣ x - 3 ∣ + \frac{1}{3} ln ∣ x ∣

Add a constant to the solution

= - \frac{2}{5} ln ∣ x + 2 ∣ + \frac{1}{15} ln ∣ x - 3 ∣ + \frac{1}{3} ln ∣ x ∣ + C

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What is the integral of (x-2)/(x(x^2-x-6)) ?
The integral of (x-2)/(x(x^2-x-6)) is -2/5 ln|x+2|+1/15 ln|x-3|+1/3 ln|x|+C