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

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

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

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

Solution steps

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

Take the partial fraction of

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

= \int \frac{3}{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{3}{x} d x - \int \frac{1}{x + 1} d x + \int \frac{1}{x - 1} d x

\int \frac{3}{x} d x = 3 ln ∣ x ∣

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

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

= 3 ln ∣ x ∣ - ln ∣ x + 1 ∣ + ln ∣ x - 1 ∣

Add a constant to the solution

= 3 ln ∣ x ∣ - ln ∣ x + 1 ∣ + ln ∣ x - 1 ∣ + C

t^{2} e^{- t} + (ln (t))^{2}

y = x^{2}, y = \frac{2}{x ^{2} + 1}

x \to 1 - lim (x^{3} - 7 + m)

f (x) = \frac{3}{2 x + 5}

y^{^{'}} = 2 y + sin (2 t)

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