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

The integral of x/((x+2)(3x-1)) is 2/7 ln|x+2|+1/21 ln|3x-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/((x+2)(3x-1))

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

\frac{2}{7} ln ∣ x + 2 ∣ + \frac{1}{21} ln ∣ 3 x - 1 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{x}{( x + 2 ) ( 3 x - 1 )} : \frac{2}{7 ( x + 2 )} + \frac{1}{7 ( 3 x - 1 )}

= \int \frac{2}{7 ( x + 2 )} + \frac{1}{7 ( 3 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{2}{7 ( x + 2 )} d x + \int \frac{1}{7 ( 3 x - 1 )} d x

\int \frac{2}{7 ( x + 2 )} d x = \frac{2}{7} ln ∣ x + 2 ∣

\int \frac{1}{7 ( 3 x - 1 )} d x = \frac{1}{21} ln ∣ 3 x - 1 ∣

= \frac{2}{7} ln ∣ x + 2 ∣ + \frac{1}{21} ln ∣ 3 x - 1 ∣

Add a constant to the solution

= \frac{2}{7} ln ∣ x + 2 ∣ + \frac{1}{21} ln ∣ 3 x - 1 ∣ + C

\int_{1}^{4} \frac{1}{x - 1} d x

\int ln (1 - x) d x

\frac{d y}{d t} = 2 (t + 1) y, y (0) = 1

\frac{d}{d x} (e^{x} - x e^{x})

\int_{1}^{2} 3 r^{4} ln (r) d r

What is the integral of x/((x+2)(3x-1)) ?
The integral of x/((x+2)(3x-1)) is 2/7 ln|x+2|+1/21 ln|3x-1|+C