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Question
`int 1/(x(x^3 - 1)) "d"x`
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Solution
Let I = `int 1/(x(x^3 - 1)) "d"x`
= `int 1/(x*x^3(1 - 1/x^3)) "d"x`
= `int 1/(x^4(1 - 1/x^3)) "d"x`
Put `1 -1/x^3` = t
Differentiating w.r.t.x, we get
`3/(x^4) "d"x` = dt
∴ `1/x^4 "d"x = 1/3 "dt"`
∴I = `1/3 int "dt"/"t"`
= `1/3 log|"t"| + "c"`
= `1/3 log|1 - 1/x^3| + "c"`
∴ I = `1/3 log |(x^3 - 1)/x^3| + "c"`
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