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Question
Resolve into partial fraction for the following:
`1/(x^2 - 1)`
Solution
Let `1/(x^2 - 1) = 1/((x + 1)(x - 1)) = "A"/(x + 1) + "B"/(x - 1)` ....(1)
`1/((x + 1)(x - 1)) = ("A"(x - 1) + "B"(x + 1))/((x + 1)(x - 1))`
1 = A(x – 1) + B(x + 1) ……. (2)
Put x = 1 in (2) we get
1 = 0 + B(1 + 1)
1 = B(2)
B = `1/2`
Put x = -1 in (2) we get
1 = A(-1 – 1) + B(-1 + 1)
1 = -2A + 0
A = `(-1)/2`
Using A = `(-1)/2`, B = `1/2` in (1) we get
`1/((x + 1)(x - 1)) = (-1/2)/(x + 1) + (1/2)/(x - 1)`
`1/(x^2 - 1) = 1/(2(x - 1)) - 1/(2(x + 1))`
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