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Question
Integrate the function `x^2/(1 - x^6)`
Solution
Let `I = x^2/(1 - x^6) dx`
`= int x^2/(1 - (x^3)^2) dx`
Put x3 = t
3x2 dx = dt ⇒ x2 dx = `1/3` dt
`therefore I = 1/3 int dt/(1 - t^2)`
`= 1/3 . 1/2 log abs ((1 + t)/(1 - t)) + C`
`= 1/6 log abs ((1 + t)/(1 - t)) + C`
`= 1/6 log abs ((1 + x^3)/(1 - x^3)) + C` `...[∵ int dx/(a^2 - x^2) = 1/(2a) log |(a + x) /(a - x)|+C]`
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