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Question
`int x^2sqrt("a"^2 - x^6) "d"x`
Solution
Let I = `int x^2sqrt("a"^2 - x^6) "d"x`
= `int sqrt("a"^2 - (x^3)^2)*x^2 "d"x`
Put x3 = t
∴ 3x2dx = dt
∴ x2dx = `1/3 "dt"`
∴ I = `1/3 int sqrt("a"^2 - "t"^2) "dt"`
= `1/3["t"/2 sqrt("a"^2 - "t"^2) + "a"^2/2 sin^-1 ("t"/"a")] + "c"`
∴ I = `1/6[x^3 sqrt("a"^2 - x^6) + "a"^2 sin^-1 (x^3/"a")] + "c"`
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