Question

Evaluate the following:

`int sqrt(x)/(sqrt("a"^3 - x^3)) "d"x`

Answer 1

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"`.