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

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

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

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

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

Solution steps

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

Expand

\frac{x + 2}{x ^{3} - x} : \frac{x}{x ^{3} - x} + \frac{2}{x ^{3} - 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{x}{x ^{3} - x} d x + \int \frac{2}{x ^{3} - x} d x

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

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

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

Simplify

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

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

Add a constant to the solution

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

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