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

The integral of x/(3x+2) is 1/9 (3x+2-2ln|3x+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/(3x+2)

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

\frac{1}{9} (3 x + 2 - 2 ln ∣ 3 x + 2 ∣) + C

Solution steps

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

Apply u-substitution

: \int \frac{u - 2}{9 u} d u

= \int \frac{u - 2}{9 u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{9} \cdot \int \frac{u - 2}{u} d u

Expand

\frac{u - 2}{u} : 1 - \frac{2}{u}

= \frac{1}{9} \cdot \int 1 - \frac{2}{u} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{9} (\int 1 d u - \int \frac{2}{u} d u)

\int 1 d u = u

\int \frac{2}{u} d u = 2 ln ∣ u ∣

= \frac{1}{9} (u - 2 ln ∣ u ∣)

Substitute back

u = 3 x + 2

= \frac{1}{9} (3 x + 2 - 2 ln ∣ 3 x + 2 ∣)

Add a constant to the solution

= \frac{1}{9} (3 x + 2 - 2 ln ∣ 3 x + 2 ∣) + C

8 x - x^{2}

\frac{d}{d x} (x^{6} sin (x))

n = 0 \sum \infty \frac{1}{10 1 ^{n}}

\frac{x}{4}, 25, 0, 12

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

What is the integral of x/(3x+2) ?
The integral of x/(3x+2) is 1/9 (3x+2-2ln|3x+2|)+C