$$\begin{gathered} \mathrm{I} = {\int }_{-\mathrm{a}}^{\mathrm{a}} \sqrt{\frac{\mathrm{a} - \mathrm{x}}{\mathrm{a} + \mathrm{x}} } \mathrm{dx} \\ = {\int }_{-\mathrm{a}}^{\mathrm{a}} \frac{\mathrm{a} - \mathrm{x}}{\sqrt{{\mathrm{a}}^2 - {\mathrm{x}}^2}} \mathrm{dx} \\ = {\int }_{-\mathrm{a}}^{\mathrm{a}} \frac{\mathrm{a}}{\sqrt{{\mathrm{a}}^2 - {\mathrm{x}}^2}} \mathrm{dx} - {\int }_{-\mathrm{a}}^{\mathrm{a}} \frac{ \mathrm{x}}{\sqrt{{\mathrm{a}}^2 - {\mathrm{x}}^2}} \mathrm{dx} \\ = {\mathrm{I}}_1 + {\mathrm{I}}_2 \\ \mathrm{Where} {\mathrm{I}}_1 ={\int }_{-\mathrm{a}}^{\mathrm{a}} \frac{\mathrm{a}}{\sqrt{{\mathrm{a}}^2 - {\mathrm{x}}^2}} \mathrm{dx}, \mathrm{which} \mathrm{is} \mathrm{the} \mathrm{integral} \mathrm{of} \mathrm{an} \mathrm{even} \mathrm{function} \\ \mathrm{And} {\mathrm{I}}_2 ={\int }_{-\mathrm{a}}^{\mathrm{a}} \frac{\mathrm{x}}{\sqrt{{\mathrm{a}}^2 - {\mathrm{x}}^2}} \mathrm{dx}, \mathrm{which} \mathrm{is} \mathrm{the} \mathrm{integral} \mathrm{of} \mathrm{an} \mathrm{even} \mathrm{function}, \\ \mathrm{and} \mathrm{so} {\mathrm{I}}_2 =0 \end{gathered}$$

Now,

$$\begin{gathered} \mathrm{I} = {\mathrm{I}}_1 = {\int }_{-\mathrm{a}}^{\mathrm{a}} \frac{\mathrm{a}}{\sqrt{{\mathrm{a}}^2 - {\mathrm{x}}^2}} \mathrm{dx} \\ = 2{\int }_0^{\mathrm{a}} \frac{\mathrm{a}}{\sqrt{{\mathrm{a}}^2 - {\mathrm{x}}^2}} \mathrm{dx} \\ = 2\mathrm{a}{\int }_0^{\mathrm{a}} \frac{1}{\sqrt{{\mathrm{a}}^2 - {\mathrm{x}}^2}} \mathrm{dx} \\ = 2\mathrm{a}. {\left[{\sin }^{-1} \left(\frac{\mathrm{x}}{\mathrm{a}}\right)\right]}_0^{\mathrm{a}} \\ = 2\mathrm{a}.\left[{\sin }^{-1} 1 - {\sin }^{-1} 0\right] \\ = 2\mathrm{a}\left(\frac{\mathrm{\pi }}{2}\right) \\ = \mathrm{\pi a} \end{gathered}$$