Question

Question #80cb5

Answer 1

`(2 sqrt(x/(a^3 - x^3)) sqrt(a^3 - x^3) tan^(-1)(x^(3/2)/sqrt(a^3 - x^3)))/(3 sqrt(x)) + C`

Explanation:

1. Simplify powers

`int sqrt(x) / sqrt(a^3 - x^3) dx`

1. Substitute u = `x^(3/2)` and du = `(3sqrt(x)) / 2`

`= 2/3 int1/sqrt(a^3 - u^2) du`

1. Take a^3 out of the root

`2/3 int1/(a^(3/2) sqrt(1 - u^2/a^3)) du`

1. Factor out contsants

`2/(3 a^(3/2)) int1/sqrt(1 - u^2/a^3) du`

1. Substitute r = `u / a^(3/2)` and dr = `a^(-3/2)` du

`2/3 int1/sqrt(1 - r^2) dr`

1. The integral of `1/sqrt(1 - r^2) is sin^(-1)(r)`

`= 2/3 sin^(-1)(s) + C`

1. Substitute back r and u

`= 2/3 sin^(-1)(x^(3/2)/a^(3/2)) + C`

1. Remove negative exponents

`(2 sqrt(x/(a^3 - x^3)) sqrt(a^3 - x^3) tan^(-1)(x^(3/2)/sqrt(a^3 - x^3)))/(3 sqrt(x)) + C`