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

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

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

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

Solution steps

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

Take the partial fraction of

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

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

\int \frac{2}{x + 1} d x = 2 ln ∣ x + 1 ∣

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

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

Add a constant to the solution

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

2 e^{- x}

\int \frac{sec ^{2} ( x )}{tan ^{3} ( x )} d x

6 t \frac{d y}{d t} + y = t

3 x^{2} - 3

x^{5} sin (x)

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