How To Integrate sqrt(1-x/1+x)

 

$$\begin{array}{l}\int \sqrt{\frac{\mathbf{1}-x}{\mathbf{1}+x}}\mathrm{d}x\\ \\ \int \sqrt{\frac{1-x}{1+x}}\ast \sqrt{\frac{1-x}{1-x}}\mathrm{d}x\\ \int \frac{1-x}{\sqrt{1-x^2}}\mathrm{d}x\\ \int \frac{1}{\sqrt{1-x^2}}\mathrm{d}x-\int \frac{x}{\sqrt{1-x^2}}\mathrm{d}x\\ {\sin }^{-1}x-\int \frac{x}{\sqrt{1-x^2}}\mathrm{d}x\\ BySubstitution\\ z=1-x^2\\ dz=-2xdx\\ {\sin }^{-1}x+\frac{1}{2}\int \frac{1}{\sqrt{z}}\mathrm{d}z\\ {\sin }^{-1}x+\frac{1}{2}\frac{{\left(z\right)}^{\frac{1}{2}}}{0.5}+c\\ \Rightarrow {\sin }^{-1}x+\sqrt{1-x^2}+c\end{array}$$