Let's rewrite the expression
1/(x(x^2-1)^2)=1/(x(x-1)^2(x+1)^2)1x(x2−1)2=1x(x−1)2(x+1)2
The decomposition into partial fractions is
1/(x(x-1)^2(x+1)^2)=A/(x)+B/(x-1)^2+C/(x-1)+D/(x+1)^2+E/(x+1)1x(x−1)2(x+1)2=Ax+B(x−1)2+Cx−1+D(x+1)2+Ex+1
=(A(x-1)^2(x+1)^2+Bx(x+1)^2+Cx(x-1)(x+1)^2+Dx(x-1)^2+Ex(x-1)^2(x+1))/(x(x-1)^2(x+1)^2)=A(x−1)2(x+1)2+Bx(x+1)2+Cx(x−1)(x+1)2+Dx(x−1)2+Ex(x−1)2(x+1)x(x−1)2(x+1)2
Therefore,
1=A(x-1)^2(x+1)^2+B(x)(x+1)^2+Cx(x-1)(x+1)^2+Dx(x-1)^2+Ex(x-1)^2(x+1)1=A(x−1)2(x+1)2+B(x)(x+1)2+Cx(x−1)(x+1)2+Dx(x−1)2+Ex(x−1)2(x+1)
Let x=0x=0, =>⇒, 1=A#
Let x=1x=1, =>⇒, 1=4B1=4B, =>⇒, B=1/4B=14
Let x=-1x=−1, =>⇒, 1=-4D1=−4D, =>⇒, D=-1/4D=−14
Coefficients of x^4x4, =>⇒,0=A+C+E0=A+C+E, =>⇒, C+E=-1C+E=−1
Coeficients of x^3x3, =>⇒, 0=B+C+D-E0=B+C+D−E, =>⇒, C-E=0C−E=0
C=E=-1/2C=E=−12
1/(x(x-1)^2(x+1)^2)=1/(x)+(1/4)/(x-1)^2+(-1/2)/(x-1)+(-1/4)/(x+1)^2+(-1/2)/(x+1)1x(x−1)2(x+1)2=1x+14(x−1)2+−12x−1+−14(x+1)2+−12x+1
