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)) + C2xa3x3a3x3tan1(x32a3x3)3x+C

Explanation:

  1. Simplify powers

int sqrt(x) / sqrt(a^3 - x^3) dxxa3x3dx

  1. Substitute u = x^(3/2)x32 and du = (3sqrt(x)) / 23x2

= 2/3 int1/sqrt(a^3 - u^2) du=231a3u2du

  1. Take a^3 out of the root

2/3 int1/(a^(3/2) sqrt(1 - u^2/a^3)) du231a321u2a3du

  1. Factor out contsants

2/(3 a^(3/2)) int1/sqrt(1 - u^2/a^3) du23a3211u2a3du

  1. Substitute r = u / a^(3/2)ua32 and dr = a^(-3/2)a32 du

2/3 int1/sqrt(1 - r^2) dr2311r2dr

  1. The integral of 1/sqrt(1 - r^2) is sin^(-1)(r)11r2issin1(r)

= 2/3 sin^(-1)(s) + C=23sin1(s)+C

  1. Substitute back r and u

= 2/3 sin^(-1)(x^(3/2)/a^(3/2)) + C=23sin1(x32a32)+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)) + C2xa3x3a3x3tan1(x32a3x3)3x+C