Question

into partial fraction
1/(X+1) (x²-1)​

Answer 1

Answer:

First write 1/((x-1)*(1+x^2) = A/(x-1) + (Bx+C)/(1+x^2). Note how the second term is written. The degree of the numerator must be one less than the degree of the denominator.

Multiply both sides by (x-1) and substitute x = 1. The second term drops out (that is why chose x = 1.) and get A = 1/2.

To get B and C, there are many ways, all of them messy. One way is to bring A/(x-1) to the left side and simplify to get

(Bx+C)/(1+x^2) = 1/(x-1) * (1/(1+x^2)-1/2) =

1/(x-1) * (1 - x^2) / (1+x^2) = -(1 + x) /(2* (1 + x^2)). Note that 1/(x-1) cancels out, as it always does. Then (Bx + C) can be identified as -(1 + x)/2 and the complete partial fraction expansion is (1/(x-1) - (1+x)/(1+x^2))/2.