What is the integral of (13x-6)/(x(3x-2)) ?

The integral of (13x-6)/(x(3x-2)) is 4/3 ln|3x-2|+3ln|x|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (13x-6)/(x(3x-2))

\int \frac{13 x - 6}{x ( 3 x - 2 )} d x

\frac{4}{3} ln ∣ 3 x - 2 ∣ + 3 ln ∣ x ∣ + C

Solution steps

\int \frac{13 x - 6}{x ( 3 x - 2 )} d x

Expand

\frac{13 x - 6}{x ( 3 x - 2 )} : \frac{13 x}{x ( 3 x - 2 )} - \frac{6}{x ( 3 x - 2 )}

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{13 x}{x ( 3 x - 2 )} d x - \int \frac{6}{x ( 3 x - 2 )} d x

\int \frac{13 x}{x ( 3 x - 2 )} d x = \frac{13}{3} ln ∣ 3 x - 2 ∣

\int \frac{6}{x ( 3 x - 2 )} d x = 6 (- \frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ 3 x - 2 ∣)

= \frac{13}{3} ln ∣ 3 x - 2 ∣ - 6 (- \frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ 3 x - 2 ∣)

Simplify

\frac{13}{3} ln ∣ 3 x - 2 ∣ - 6 (- \frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ 3 x - 2 ∣) : \frac{4}{3} ln ∣ 3 x - 2 ∣ + 3 ln ∣ x ∣

= \frac{4}{3} ln ∣ 3 x - 2 ∣ + 3 ln ∣ x ∣

Add a constant to the solution

= \frac{4}{3} ln ∣ 3 x - 2 ∣ + 3 ln ∣ x ∣ + C

n = 1 \sum \infty \frac{( 1 ^{n} )}{n}

\frac{d}{d x} ((2 x) d x)

\int \frac{ln ( y )}{y} d y

\frac{d}{d x} (x^{2} e^{2})

y^{^{''}} - y^{^{'}} - 6 y = e^{(- x)}

What is the integral of (13x-6)/(x(3x-2)) ?
The integral of (13x-6)/(x(3x-2)) is 4/3 ln|3x-2|+3ln|x|+C