What is the integral of (x^2+12x-12)/(x^3-4x) ?

The integral of (x^2+12x-12)/(x^3-4x) is 3ln|x|-4ln|x+2|+2ln|x-2|+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+12x-12)/(x^3-4x)

\int \frac{x ^{2} + 12 x - 12}{x ^{3} - 4 x} d x

3 ln ∣ x ∣ - 4 ln ∣ x + 2 ∣ + 2 ln ∣ x - 2 ∣ + C

Solution steps

\int \frac{x ^{2} + 12 x - 12}{x ^{3} - 4 x} d x

Take the partial fraction of

\frac{x ^{2} + 12 x - 12}{x ^{3} - 4 x} : \frac{3}{x} - \frac{4}{x + 2} + \frac{2}{x - 2}

= \int \frac{3}{x} - \frac{4}{x + 2} + \frac{2}{x - 2} 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{4}{x + 2} d x + \int \frac{2}{x - 2} d x

\int \frac{3}{x} d x = 3 ln ∣ x ∣

\int \frac{4}{x + 2} d x = 4 ln ∣ x + 2 ∣

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

= 3 ln ∣ x ∣ - 4 ln ∣ x + 2 ∣ + 2 ln ∣ x - 2 ∣

Add a constant to the solution

= 3 ln ∣ x ∣ - 4 ln ∣ x + 2 ∣ + 2 ln ∣ x - 2 ∣ + C

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

\int tan^{3} (x) sin^{2} (x) d x

\int (t^{3} - \frac{e ^{- t} + 4}{e ^{- t}}) d t

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

\int \frac{sin ( 2 x )}{( 1 - cos ( 2 x ) ) ^{2}} d x

What is the integral of (x^2+12x-12)/(x^3-4x) ?
The integral of (x^2+12x-12)/(x^3-4x) is 3ln|x|-4ln|x+2|+2ln|x-2|+C