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Question
Integrate the function:
`1/(x - x^3)`
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Solution
Let `1/(x - x^3) = 1/(x(1 + x)(1 - x))`
`≡ A/x + B/(1 + x) + C/(1 - x)`
⇒ 1 = A (1 + x) (1 – x) + Bx (1 – x) + Cx (1 + x) …(1)
Putting x = 0 in equation (1),
1 = A(1 + 0) (1 – 0)
⇒ A = 1
Putting x = -1 in equation (1),
1 = B (-1) (1 + 1)
`=> B = - 1/2`
Putting x = 1 in equation (1),
1 = C(1)(1 + 1)
`=> C = 1/2`
`therefore 1/(x - x^3) = 1/x - 1/(2(1 + x)) + 1/(2(1 - x))`
`therefore int 1/(x - x^3) dx = int 1/x dx - 1/2 int 1/(1 + x) dx + 1/2 int 1/(1 - x) dx`
`= log |x| - 1/2 log |1 + x| - 1/2 log |1 - x| + C`
`= 1/2 log |x|^2 - 1/2 log |1 - x^2| + C`
`= 1/2 log |x^2/(1 - x^2)|` + C
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