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Question
Find : `int sqrt(x/(1 - x^3))dx; x ∈ (0, 1)`.
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Solution
Let `x^(3/2)` = t
`\implies` dt = `3/2 x^(1/2) dx`
`int sqrt(x/(1 - x^3))dx = 2/3 int dt/sqrt(1 - t^2)`
= `2/3 sin^-1 (t) + c`
= `2/3 sin^-1 (x^(3/2)) + c`, where 'c' is an arbitrary constant of integration.
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