What is the integral of (x+2)/(x^4-x^2) ?

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

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integral of (x+2)/(x^4-x^2)

\int \frac{x + 2}{x ^{4} - x ^{2}} d x

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

Solution steps

\int \frac{x + 2}{x ^{4} - x ^{2}} d x

Apply Integration By Parts

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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