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

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

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

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

Solution steps

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

Apply u-substitution

: \int \frac{u + 2}{u ^{3}} d u

= \int \frac{u + 2}{u ^{3}} d u

Expand

\frac{u + 2}{u ^{3}} : \frac{1}{u ^{2}} + \frac{2}{u ^{3}}

= \int \frac{1}{u ^{2}} + \frac{2}{u ^{3}} 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

= \int \frac{1}{u ^{2}} d u + \int \frac{2}{u ^{3}} d u

\int \frac{1}{u ^{2}} d u = - \frac{1}{u}

\int \frac{2}{u ^{3}} d u = - \frac{1}{u ^{2}}

= - \frac{1}{u} - \frac{1}{u ^{2}}

Substitute back

u = x - 2

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

Add a constant to the solution

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

\frac{d}{d x} (\frac{1}{4} e^{x} + \frac{3}{4} e^{- x})

\frac{d}{d x} (5 ln (x) - \frac{x ^{2}}{40})

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

x \to 4 lim (\frac{6}{x})

\int x y^{2} d x

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