First, note that we can factor the denominator as
x^4 - 1 = (x^2+1)(x^2-1)=(x^2+1)(x+1)(x-2)x4−1=(x2+1)(x2−1)=(x2+1)(x+1)(x−2)
Then, using partial fraction decomposition,
(x^3-2)/(x^4-1) = (Ax+B)/(x^2+1)+C/(x+1)+D/(x-1)x3−2x4−1=Ax+Bx2+1+Cx+1+Dx−1
Multiplying through by (x^2+1)(x+1)(x-1)(x2+1)(x+1)(x−1) gives
x^3-2x3−2
= (Ax+B)(x^2-1) + C(x^2+1)(x-1) + D(x^2+1)(x+1)=(Ax+B)(x2−1)+C(x2+1)(x−1)+D(x2+1)(x+1)
= (A + C + D)x^3 + (B - C + D)x^2 + (-A + C + D)x + (-B -C+D)=(A+C+D)x3+(B−C+D)x2+(−A+C+D)x+(−B−C+D)
Equating coefficients then gives us the system
{(A+C+D = 1), (B-C+D = 0), (-A + C + D = 0), (-B-C+D = -2):}
Solving this gives us
{(A = 1/2), (B = 1), (C = 3/4), (D = -1/4):}
Substituting back, we get the final result.
(x^3-2)/(x^4-1) = (1/2x+1)/(x^2+1)+(3/4)/(x+1)+(-1/4)/(x-1)
= (x+2)/(2(x^2+1))+3/(4(x+1))-1/(4(x-1))
