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

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

Popular >

integral of x/((x-2)^2)

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

ln ∣ x - 2 ∣ - \frac{2}{x - 2} + C

Solution steps

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

Apply u-substitution

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

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

Expand

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

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

\int \frac{1}{u} d u = ln ∣ u ∣

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

= ln ∣ u ∣ - \frac{2}{u}

Substitute back

u = x - 2

= ln ∣ x - 2 ∣ - \frac{2}{x - 2}

Add a constant to the solution

= ln ∣ x - 2 ∣ - \frac{2}{x - 2} + C

x \to - \infty lim (x - 1)

\frac{\partial ^{2}}{\partial x ^{2}} (x y^{2} - 6 x^{2} - 3 y^{2})

\frac{2 x ^{2} + x - 3}{- 3 x ^{2} - 4 x + 2}

(e^{- x} - e^{5}) (e^{x} + e^{- 3})

\frac{400}{s ( s + 2 ) ^{3}}

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