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

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

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

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

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

Solution steps

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

Take the partial fraction of

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

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

\int 1 d x = x

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

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

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

Add a constant to the solution

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

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