Question

Resolve `(2x-3)/((x-1)(x^(2)+1)^(2))` into partial fractions,

Answer 1

Let `(2x-3)/((x-1)(x^(2)+1)^(2))=A/(x-1)+(Bx+C)/(x^(2)+1)+(Dx+E)/(x^(2)+1)^(2)`. Then, `2x-3=A(x^(2)+1)^(2)+(Bx+C)(x-1)(x^(2)+1)+(Dx+E)(x-1)`…………(i)
Putting x=1 in eq (i) , we get `-1=A(1+1)^(2) rArr A=-`
Comparing coefficients of like powers of x on both side of (i), we have
`A+B=0, C-B=0, 2A+B-C+D=0, C+E-B-D=2` and `A-C-E=-3`.
Putting `A=-1/4` and solving these equations, we get
`B=1/4=C,D=1/4` and `E=5/2 therefore (2x-3)/((x-1)(x^(2)+1)^(2))=(-1)/(4(x-1))+(x+1)/(4(x^(2)+1))+(9x+5)/(2(x^(2)+1)^(2))`