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Question
Evaluate the following:
`int sqrt(x)/(sqrt("a"^3 - x^3)) "d"x`
Solution
Let I = `int sqrt(x)/(sqrt("a"^3 - x^3)) "d"x`
= `int x^(3/2)/sqrt(("a"^(3/2))^2 - (x^(3/2))^2) "d"x`
Put `x^(3/2)` = t
⇒ `3/2 x^(1/2) "d"x` = dt
⇒ `x^(1/2) "d"x = 2/3 "dt"`
∴ I = `2/3 int "dt"/sqrt(("a"^(3/2))^2 - ("t")^2)`
= `2/3 sin^-1 "t"/("a"^(3/2)) + "C"`
= `2/3 sin^-1 ((x^(3/2))/("a"^(3/2))) + "C"`
Hence I = `2/3 sin^-1 (x/"a")^(3/2) + "C"`.
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