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

The integral of (x-2)/(x^5-x) is -1/2 arctan(x)-3/4 ln|x+1|-1/4 ln|x-1|-1/2 ln|x^2+1|+2ln|x|+C

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

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

- \frac{1}{2} arctan (x) - \frac{3}{4} ln ∣ x + 1 ∣ - \frac{1}{4} ln ∣ x - 1 ∣ - \frac{1}{2} ln x^{2} + 1 + 2 ln ∣ x ∣ + C

Solution steps

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

Expand

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

\int \frac{x}{x ^{5} - x} d x = - \frac{1}{2} arctan (x) - \frac{1}{4} ln ∣ x + 1 ∣ + \frac{1}{4} ln ∣ x - 1 ∣

\int \frac{2}{x ^{5} - x} d x = 2 (- ln ∣ x ∣ + \frac{1}{4} ln x^{2} + 1 + \frac{1}{4} ln ∣ x + 1 ∣ + \frac{1}{4} ln ∣ x - 1 ∣)

= - \frac{1}{2} arctan (x) - \frac{1}{4} ln ∣ x + 1 ∣ + \frac{1}{4} ln ∣ x - 1 ∣ - 2 (- ln ∣ x ∣ + \frac{1}{4} ln x^{2} + 1 + \frac{1}{4} ln ∣ x + 1 ∣ + \frac{1}{4} ln ∣ x - 1 ∣)

Simplify

- \frac{1}{2} arctan (x) - \frac{1}{4} ln ∣ x + 1 ∣ + \frac{1}{4} ln ∣ x - 1 ∣ - 2 (- ln ∣ x ∣ + \frac{1}{4} ln x^{2} + 1 + \frac{1}{4} ln ∣ x + 1 ∣ + \frac{1}{4} ln ∣ x - 1 ∣) : - \frac{1}{2} arctan (x) - \frac{3}{4} ln ∣ x + 1 ∣ - \frac{1}{4} ln ∣ x - 1 ∣ - \frac{1}{2} ln x^{2} + 1 + 2 ln ∣ x ∣

= - \frac{1}{2} arctan (x) - \frac{3}{4} ln ∣ x + 1 ∣ - \frac{1}{4} ln ∣ x - 1 ∣ - \frac{1}{2} ln x^{2} + 1 + 2 ln ∣ x ∣

Add a constant to the solution

= - \frac{1}{2} arctan (x) - \frac{3}{4} ln ∣ x + 1 ∣ - \frac{1}{4} ln ∣ x - 1 ∣ - \frac{1}{2} ln x^{2} + 1 + 2 ln ∣ x ∣ + C

\int (2 x^{- 3} + 3 x^{- 2} + 2) d x

\int sinh (3 x - 1) d x

\int cos^{3} (\frac{x}{4}) d x

\int (3 x^{2} - 3 y^{2}) d y

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

What is the integral of (x-2)/(x^5-x) ?
The integral of (x-2)/(x^5-x) is -1/2 arctan(x)-3/4 ln|x+1|-1/4 ln|x-1|-1/2 ln|x^2+1|+2ln|x|+C