# How do you find the indefinite integral of int (x(x-2))/(x-1)^3?

Dec 19, 2016

Substitute $u = x - 1$ to get $\ln | x - 1 | + \frac{1}{2 {\left(x - 1\right)}^{2}} + C$

#### Explanation:

$\int \frac{x \left(x - 2\right)}{x - 1} ^ 3 \mathrm{dx}$
Substitute $u = x - 1$, $\mathrm{dx} = \mathrm{du}$:
$= \int \frac{\left(u + 1\right) \left(u - 1\right)}{u} ^ 3 \mathrm{du}$
$= \int {u}^{-} 1 - {u}^{-} 3 \mathrm{dx}$
$= \ln | u | + \frac{1}{2} {u}^{-} 2 + C$
$= \ln | x - 1 | + \frac{1}{2 {\left(x - 1\right)}^{2}} + C$

Whenever the denominator of the rational function is just a (positive) power of a (linear?) expression this will work, because it "pushes all the messy stuff upstairs".