after subst √x=sinθ
the integral reduces to ∫ 2/(1+sinθ)
=∫ sec2 θ/2 / (1+tanθ/2)2
now put tan θ/2 =t and proceed
dx/(1+√x)(√x)(√(1-x))
√x=sinθ
dx/2√x=cosθdθ
∫2cosθdθ/(1+sinθ)(cosθ)
=∫2dθ/(1+sinθ)
after subst √x=sinθ
the integral reduces to ∫ 2/(1+sinθ)
=∫ sec2 θ/2 / (1+tanθ/2)2
now put tan θ/2 =t and proceed