Question #80cb5

1 Answer
Apr 8, 2017

#(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#