Question

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

Answer 1

The answer is `=ln(x-2)+3/(x-1)+C`

Explanation:

Let's start with the decomposition of the partial fractions
`=(A+B(x-2))/(x-2)^2`
`:.` `x-5=A+B(x-2)`
Coefficients of x `B=1`
and `-5=A-2B` `=>` `A=-3`
`(x-5)/(x-2)^2=-3/(x-2)^2+1/(x-2)`
So `int((x-5)dx)/(x-2)^2=intdx/(x-2)-int(3dx)/(x-2)^2`
`=ln(x-2)+3/(x-1)+C`