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Question
Integrate the functions:
`(x^3 - 1)^(1/3) x^5`
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Solution
Let `I = int (x^3 - 1)^(1/3) .x^5 dx`
On multiplying the numerator and denominator by 3
`I = 1/3 int (x^3 - 1)^(1/3).3x^2 . x^3` dx
Put x3 - 1 = t
3x2 dx = dt
Also, x3 = t + 1
∴ `I = 1/3 int t^(1/3) (t + 1) dt`
`= 1/3 [3/7 t ^(7/3) + 3/4 t^(4/3)] + C = 1/7 t^(7/3) + 1/4 t^(4/3) C`
`= 1/7 (x^3 - 1)^(7/3) + 1/4 (x^3 - 1)^(4/3) + C`
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