What is the integral of (2-4x^2)/(x-x^3) ?

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

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integral of (2-4x^2)/(x-x^3)

\int \frac{2 - 4 x ^{2}}{x - x ^{3}} d x

2 ln ∣ x ∣ + ln ∣ x + 1 ∣ + ln ∣ x - 1 ∣ + C

Solution steps

\int \frac{2 - 4 x ^{2}}{x - x ^{3}} d x

Expand

\frac{2 - 4 x ^{2}}{x - x ^{3}} : \frac{2}{x - x ^{3}} - \frac{4 x ^{2}}{x - x ^{3}}

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{2}{x - x ^{3}} d x - \int \frac{4 x ^{2}}{x - x ^{3}} d x

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

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

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

Simplify

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

= 2 ln ∣ x ∣ + ln ∣ x + 1 ∣ + ln ∣ x - 1 ∣

Add a constant to the solution

= 2 ln ∣ x ∣ + ln ∣ x + 1 ∣ + ln ∣ x - 1 ∣ + C

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