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Question
Integrate the rational function:
`1/(x(x^4 - 1))`
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Solution
Let `I = int 1/ (x (x^4 - 1)) dx`
`= 1/4 int (4x^3)/(x^4(x^4 - 1)) dx`
Put x4 = t
⇒ 4x3 dx = dt
∴ `I = 1/4 int dt/(t(t - 1))`
Let `1/(t (t - 1)) = A/t + B/(t - 1)`
⇒ 1 = A (t - 1) + Bt ....(i)
Putting t = 0 in (i), we get
1 = A (-1)
⇒ A = -1
Putting t = 1 in (i), we get
1 = B (1)
⇒ B = 1
∴ `1/ (t (t - 1)) = (-1)/t + 1/ (t - 1)`
∴ `I = 1/4 int (-1/t + 1/ (t - 1)) dt`
`= 1/4 [-log |t| + log |t - 1|] + C`
`= 1/4 log |(t - 1)/t| + C`
`= 1/4 log |(x^4 - 1)/x^4| + C`
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