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

The integral of (3x+2)/(x^2-x-2) is 1/3 ln|x+1|+8/3 ln|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 (3x+2)/(x^2-x-2)

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

\frac{1}{3} ln ∣ x + 1 ∣ + \frac{8}{3} ln ∣ x - 2 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{3 x + 2}{x ^{2} - x - 2} : \frac{1}{3 ( x + 1 )} + \frac{8}{3 ( x - 2 )}

= \int \frac{1}{3 ( x + 1 )} + \frac{8}{3 ( 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{1}{3 ( x + 1 )} d x + \int \frac{8}{3 ( x - 2 )} d x

\int \frac{1}{3 ( x + 1 )} d x = \frac{1}{3} ln ∣ x + 1 ∣

\int \frac{8}{3 ( x - 2 )} d x = \frac{8}{3} ln ∣ x - 2 ∣

= \frac{1}{3} ln ∣ x + 1 ∣ + \frac{8}{3} ln ∣ x - 2 ∣

Add a constant to the solution

= \frac{1}{3} ln ∣ x + 1 ∣ + \frac{8}{3} ln ∣ x - 2 ∣ + C

\int \frac{3 e ^{x}}{e ^{2 x} + 9 e ^{x} + 20} d x

\int 6 y e^{5 y^{2}} d y

\int z sin (z) cos (z) d z

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

\int (x^{3} + 4) d x

What is the integral of (3x+2)/(x^2-x-2) ?
The integral of (3x+2)/(x^2-x-2) is 1/3 ln|x+1|+8/3 ln|x-2|+C