IIT JEE , AIEEE Forum - Mathematics , Integral Calculus - integrate the following function give full explanation

ramkumar_november

the given integral is :::

I = $\int \sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}\,dx$

put $\sqrt{x}=sin\theta$

I = $\int \sqrt{\frac{1-sin\theta}{1+sin\theta}} \;sin2\theta\,d\theta$

I = $\int \sqrt{\frac{(1-sin\theta)(1+sin\theta)}{(1+sin\theta)^2}} \;sin2\theta\,d\theta$

I = $\int \frac{2sin\theta cos^2\theta\,d\theta}{1+sin\theta}$

I = $\int \frac{2sin\theta(1- sin^2\theta)\,d\theta}{1+sin\theta}$

I = $\int 2sin\theta(1-sin\theta)\,d\theta$

I = $2\int sin\theta\,d\theta$ - $2\int sin^2\theta\,d\theta$

I = $-2cos\theta\;-\;2\int \frac{1-cos2\theta}{2}\;d\theta$

I = $-2cos\theta\;-\;\theta\;+\frac{sin2\theta}{2}$

where $\theta\;=\;sin^{-1}\sqrt{x}$

clear with my proof??