What is the integral of (2x^2-24)/(x(x+2)^3) ?

The integral of (2x^2-24)/(x(x+2)^3) is -8/(x+2)-4/((x+2)^2)-3ln(1/2 |x|)+3ln| x/2+1|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (2x^2-24)/(x(x+2)^3)

\int \frac{2 x ^{2} - 24}{x ( x + 2 ) ^{3}} d x

- \frac{8}{x + 2} - \frac{4}{( x + 2 ) ^{2}} - 3 ln (\frac{1}{2} ∣ x ∣) + 3 ln \frac{x}{2} + 1 + C

Solution steps

\int \frac{2 x ^{2} - 24}{x ( x + 2 ) ^{3}} d x

Expand

\frac{2 x ^{2} - 24}{x ( x + 2 ) ^{3}} : \frac{2 x}{( x + 2 ) ^{3}} - \frac{24}{x ( x + 2 ) ^{3}}

= \int \frac{2 x}{( x + 2 ) ^{3}} - \frac{24}{x ( x + 2 ) ^{3}} 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}{( x + 2 ) ^{3}} d x - \int \frac{24}{x ( x + 2 ) ^{3}} d x

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

\int \frac{24}{x ( x + 2 ) ^{3}} d x = 6 (\frac{1}{x + 2} + \frac{1}{( x + 2 ) ^{2}} + \frac{1}{2} ln (\frac{1}{2} ∣ x ∣) - \frac{1}{2} ln \frac{x}{2} + 1)

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

Simplify

2 (- \frac{1}{x + 2} + \frac{1}{( x + 2 ) ^{2}}) - 6 (\frac{1}{x + 2} + \frac{1}{( x + 2 ) ^{2}} + \frac{1}{2} ln (\frac{1}{2} ∣ x ∣) - \frac{1}{2} ln \frac{x}{2} + 1) : - \frac{8}{x + 2} - \frac{4}{( x + 2 ) ^{2}} - 3 ln (\frac{1}{2} ∣ x ∣) + 3 ln \frac{x}{2} + 1

= - \frac{8}{x + 2} - \frac{4}{( x + 2 ) ^{2}} - 3 ln (\frac{1}{2} ∣ x ∣) + 3 ln \frac{x}{2} + 1

Add a constant to the solution

= - \frac{8}{x + 2} - \frac{4}{( x + 2 ) ^{2}} - 3 ln (\frac{1}{2} ∣ x ∣) + 3 ln \frac{x}{2} + 1 + C

\int \frac{( x - 2 ) ^{n}}{n} d x

\frac{d}{d x} (\frac{x}{2 x - 7})

\int_{9}^{25} \frac{x - 9}{x} d x

(cos (x) + y sin (x)) d x + (cos (x)) d y = 0

\frac{\partial}{\partial x} (6 x y e^{- x^{2} - y^{2}})

What is the integral of (2x^2-24)/(x(x+2)^3) ?
The integral of (2x^2-24)/(x(x+2)^3) is -8/(x+2)-4/((x+2)^2)-3ln(1/2 |x|)+3ln| x/2+1|+C