\[\begin{array}{l} \smallint \sqrt {\frac{{{\bf{1}} - x}}{{{\bf{1}} + x}}} {\rm{d}}x\\\\ \smallint \sqrt {\frac{{1 - x}}{{1 + x}}} * \sqrt {\frac{{1 - x}}{{1 - x}}} {\rm{d}}x\\ \smallint \frac{{1 - x}}{{\sqrt {1 - {x^2}} }}{\rm{d}}x\\ \smallint \frac{1}{{\sqrt {1 - {x^2}} }}{\rm{d}}x - \smallint \frac{x}{{\sqrt {1 - {x^2}} }}{\rm{d}}x\\ {\sin ^{ - 1}}x - \smallint \frac{x}{{\sqrt {1 - {x^2}} }}{\rm{d}}x\\ By\;Substitution\;\;\;\\ z = 1 - {x^2}\;\;\\ \;\;\;\;\;\;dz = - 2x\;dx\\ {\sin ^{ - 1}}x + \frac{1}{2}\smallint \frac{1}{{\sqrt z }}{\rm{d}}z\\ {\sin ^{ - 1}}x + \frac{1}{2}\frac{{{{\left( z \right)}^{\frac{1}{2}}}}}{{0.5}} + c\\ \Rightarrow {\sin ^{ - 1}}x + \sqrt {1 - {x^2}} + c \end{array}\]
Tags:
integration