Question

Use partial fraction to evaluate the following:

(-2x+4)/(x^2+1)(x-1)^2

Answer 1

I will assume your expression is
(-2x+4)/( (x^2+1)(x-1)^2 )

let (-2x+4)/( (x^2+1)(x-1)^2 )
= A/(x-1) + (Bx+C)/(x-1)^2 + (Dx + E)/(x^2 + 1)

then A(x-1)(x^2+1) + (Bx+C)(x^2 + 1) + (Dx+E)(x-1)^2 = -2x + 4
let x = 1
0 + 2B + 2C + 0 = 2 ------> B + C = 1
let x = 0
-A + C + E = 4 ----------> A - C - E = -1
let x = -1
-4A - 2B + 2C - 4D + 4E = 6 ----> 2A + B - C + 2D - 2E = -3
let x = 2
5A + 10B + 5C + 2D + E = 0 ---? 5A + 10B + 5C + 2D + E = 0
let x = -2
-15A - 10B - 5C - 18D + 9E = 8

I ran these 5 equations through a "equation solving" app and got
A = 3/10, B = -4/5, C = 9/5, D = -1, and E = -1/2

take these and form the fractions.

Answer 2

Thank you!