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

The integral of x/(2x^2-3x-2) is 1/10 ln|2x+1|+2/5 ln|x-2|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

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

\frac{1}{10} ln ∣ 2 x + 1 ∣ + \frac{2}{5} ln ∣ x - 2 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{x}{2 x ^{2} - 3 x - 2} : \frac{1}{5 ( 2 x + 1 )} + \frac{2}{5 ( x - 2 )}

= \int \frac{1}{5 ( 2 x + 1 )} + \frac{2}{5 ( 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}{5 ( 2 x + 1 )} d x + \int \frac{2}{5 ( x - 2 )} d x

\int \frac{1}{5 ( 2 x + 1 )} d x = \frac{1}{10} ln ∣ 2 x + 1 ∣

\int \frac{2}{5 ( x - 2 )} d x = \frac{2}{5} ln ∣ x - 2 ∣

= \frac{1}{10} ln ∣ 2 x + 1 ∣ + \frac{2}{5} ln ∣ x - 2 ∣

Add a constant to the solution

= \frac{1}{10} ln ∣ 2 x + 1 ∣ + \frac{2}{5} ln ∣ x - 2 ∣ + C

What is the integral of x/(2x^2-3x-2) ?
The integral of x/(2x^2-3x-2) is 1/10 ln|2x+1|+2/5 ln|x-2|+C