\begin {align*} \left ( y^{\prime }\right ) ^{2} & =\frac {1-y^{2}}{1-x^{2}}\\ \frac {\left ( y^{\prime }\right ) ^{2}}{1-y^{2}} & =\frac {1}{1-x^{2}}\\ \left ( \frac {y^{\prime }}{\left ( 1-y^{2}\right ) ^{\frac {1}{2}}}\right ) ^{2} & =\frac {1}{1-x^{2}} \end {align*}
Hence we have 2 solutions\begin {align*} \frac {y^{\prime }}{\sqrt {\left ( 1-y^{2}\right ) }} & =\left \{ \begin {array} [c]{c}\sqrt {\frac {1}{1-x^{2}}}\\ -\sqrt {\frac {1}{1-x^{2}}}\end {array} \right . \\ \int \frac {dy}{\sqrt {\left ( 1-y^{2}\right ) }} & =\left \{ \begin {array} [c]{c}\int \sqrt {\frac {1}{1-x^{2}}}dx\\ -\int \sqrt {\frac {1}{1-x^{2}}}dx \end {array} \right . \\ & =\left \{ \begin {array} [c]{c}\int \frac {1}{\sqrt {1-x^{2}}}dx\\ -\int \frac {1}{\sqrt {1-x^{2}}}dx \end {array} \right . \qquad -1