Use the substitution x = sin² θ to show that ∫ √[x/(1-x)] dx = ∫ 2sin² θ dθ.?

Answer 1

∫ √[x/(1-x)]dx

x = sin² θ
dx = 2sinθcosθdθ

∫ √[sin² θ/(1-sin² θ)] 2sinθcosθ dθ
∫ √[sin² θ/(cos² θ)] 2sinθcosθ dθ
∫ [sinθ/cos θ] 2sinθcosθ dθ
∫ 2sin² θdθ

Answer 2

x= sin ² θ
dx = 2 sin θ cos θ dθ
I = ∫ √ [ sin ² θ / (1 - sin ² θ)] 2 sin θ cos θ dθ
I = ∫ [sin θ / cos θ ] 2 sinθ cos θ dθ
I = ∫ 2 sin ² θ dθ
(as required)