Question

How do you use partial fraction decomposition to decompose the fraction to integrate `(x^2-x+2)/(x(x-1)^2)`?

Answer 1

`(x^2-x+2)/(x(x-1)^2) = 2/x-1/(x-1)+2/(x-1)^2`

Explanation:

`(x^2-x+2)/(x(x-1)^2) = A/x+B/(x-1)+C/(x-1)^2`

Multiply both sides by `x(x-1)^2` (or get a common denominator and compare the numerators -- you'll get to the same place, but one approach may be clearer to you)

`A(x-1)^2+Bx(x-1)+Cx = x^2-x+2`

`Ax^2-2Ax+1+Bx^2-Bx+Cx = x^2-x+2`

`[A+B]x^2 + [-2A-B+C]x +[A] = 1x^2-1x+2`

So we get:

`A+B = 1`
`-2A+B+C=-1`
`A = 2`

Eq 1 and 3 get us `A=2` and `B = -1`

Substituting in Eq 2, we get `C = 2`