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Question
Integrate the following functions w.r.t. x : `x^2 .sqrt(a^2 - x^6)`
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Solution
Let I = `int x^2 .sqrt(a^2 - x^6).dx`
Put x3 = t
∴ 3x2.dx = dt
∴ x2dx = `(1)/(3).dt`
∴ I = `int sqrt(a^2 - t^2).dt/(3) = (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`
= `(1)/(6)[x^3 sqrt(a^2 - x^6) + a^2sin^-1 (x^3/a)] + c`.
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