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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of 2/(x^3-x)

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

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

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{1}{x ^{3} - x} d x

Take the partial fraction of

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

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

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

Simplify

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