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Question
Resolve the following rational expressions into partial fractions
`1/(x^2 - "a"^2)`
Solution
`1/(x^2 - "a"^2) = 1/((x + "a")(x - "a"))`
`1/(x^2 - "a"^2) = "A"/(x + "a") + "B"/(x - "a")`
`1/(x^2 - "a"^2) = ("A"(x - "a") + "B"(x + "a"))/((x + "a")(x - "a"))`
1 = A(x – a) + B(x + a) ......(1)
Put x = a in equation (1)
1 = A(0) + B(a + a)
1 = B(2a)
⇒ B = `1/(2"a")`
Put x = – a in equation (1)
1 = A(– a – a) + B(– a + a)
1 = – 2a A + 0
⇒ A = `- 1/(2"a")`
∴ The required partial fraction is
`1/(x^2 - "a"^2) = (-1/(2"a"))/(x + "a") + (1/(2"a"))/(x -- "a")`
`1/(x^2 - "a"^2) = 1/(2"a"(x - "a")) - 1/(2"a"(x + "a"))`
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