Question

How do you integrate `(x-5)/(x-2)^2` using partial fractions?

Answer 1

`ln|x-2|+3/(x-2)+C`

Explanation:

`int(x-5)/(x-2)^2dx`
`=int(x-2-3)/(x-2)^2dx`
`=int(cancel(x-2)/(x-2)^cancel2-3/(x-2)^2)dx`
`=int(1/(x-2)-3/(x-2)^2)dx`
`=ln|x-2|+3/(x-2)+C`.

If the last step is not obvious, substitute `u=x-2`. Alternatively, substitute `u=x-2` right at the start to get `int(1/u-3/u^2)du`, but this isn't then really partial fractions. However, it isn't really a partial fractions question because you have only one unique term in the denominator.