Question

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

Answer 1

Substitute `u=x-1` to get `ln|x-1|+1/(2(x-1)^2)+C`

Explanation:

`int (x(x-2))/(x-1)^3dx`
Substitute `u=x-1`, `dx=du`:
`=int((u+1)(u-1))/u^3du`
`=int u^-1-u^-3dx`
`=ln|u|+1/2u^-2+C`
`=ln|x-1|+1/(2(x-1)^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".