Question

Question #85073

Answer 1

`(x^2+1)/(x(x-1)^3) = -1/x + 1/(x-1)+2/(x-1)^3`

Explanation:

Performing the decomposition, we first set up the equation with the unknown constants.

`(x^2+1)/(x(x-1)^3) = A/x + B/(x-1) + C/(x-1)^2 + D/(x-1)^3`

Multiplying both sides by `x(x-1)^3`, we get

`x^2+1 = A(x-1)^3 + Bx(x-1)^2 + Cx(x-1) + Dx`

`=(A+B)x^3+(-3A-2B+C)x^2+(3A+B-C+D)x+(-A)`

Equating the coefficients of corresponding powers of `x`, we end up with the following system of equations:

`{(A+B=0),(-3A-2B+C=1),(3A+B-C+D=0),(-A=1):}`

From the fourth equation, we have `A=-1`

Substituting that into the first equation and solving for `B` gives `B=1`

Substituting both of those into the second equation and solving for `C` gives `C = 0`

Substituting all of those into the third equation and solving for `D` gives `D = 2`

Thus, from our original equation, we have the decomposition

`(x^2+1)/(x(x-1)^3) = -1/x + 1/(x-1)+2/(x-1)^3`