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

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

Popular >

integral of ((x^2+2x-1))/((x^3-x))

integral

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

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

Solution steps

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

Take the partial fraction of

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

= \int \frac{1}{x} - \frac{1}{x + 1} + \frac{1}{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 \frac{1}{x} d x - \int \frac{1}{x + 1} d x + \int \frac{1}{x - 1} d x

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

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

Add a constant to the solution

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

ln (\frac{1}{x})

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

\frac{( e ^{x} + e ^{- x} )}{2}

\frac{2 x}{( x ^{2} + 1 ) ^{2}}

y^{2} x^{3}

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