I = $\underset{-\mathit{a}}{\overset{\mathit{a}}{\int }}\sqrt{\frac{\mathit{a}-\mathit{x}}{\mathit{a}+\mathit{x}}}$ dx
= $\underset{-\mathit{a}}{\overset{\mathit{a}}{\int }}$ $\frac{\mathit{a}-\mathit{x}}{\sqrt{{\mathit{a}}^2-{\mathit{x}}^2}}$ dx
$\underset{-\mathit{a}}{\overset{\mathit{a}}{\int }}$ $\frac{\mathit{a}}{\sqrt{{\mathit{a}}^2-{\mathit{x}}^2}}\mathit{d}\mathit{x}$ - $\underset{-\mathit{a}}{\overset{\mathit{a}}{\int }}$ $\frac{\mathit{x}}{\sqrt{{\mathit{a}}^2-{\mathit{x}}^2}}$ dx
= ${\mathit{I}}_1+{\mathit{I}}_2$
Where ${\mathit{I}}_1$ = $\underset{-\mathit{a}}{\overset{\mathit{a}}{\int }}\frac{\mathit{a}}{\sqrt{{\mathit{a}}^2-{\mathit{x}}^2}}$ dx , which is the integral of an even function
And ${\mathit{I}}_2$ = $\underset{-\mathit{a}}{\overset{\mathit{a}}{\int }}$ $\frac{\mathit{x}}{\sqrt{{\mathit{a}}^2-{\mathit{x}}^2}}$ , which is the integral of an odd function, and so ${\mathit{I}}_2$ = 0
Now, I = ${\mathit{I}}_1$ = $\underset{-\mathit{a}}{\overset{\mathit{a}}{\int }}\frac{\mathit{a}}{\sqrt{{\mathit{a}}^2-{\mathit{x}}^2}}$ dx
= 2 $\underset{0}{\overset{\mathit{a}}{\int }}\frac{\mathit{a}}{\sqrt{{\mathit{a}}^2-{\mathit{x}}^2}}$ dx
2a $\underset{0}{\overset{\mathit{a}}{\int }}\frac{1}{\sqrt{{\mathit{a}}^2-{\mathit{x}}^2}}$ dx
= 2a ${\left|\mathit{s}\mathit{i}{\mathit{n}}^{-1}\left(\frac{\mathit{x}}{\mathit{a}}\right)\right|}_0^{\mathit{a}}$
= 2a $\left|\mathit{s}\mathit{i}{\mathit{n}}^{-1}1-\mathit{s}\mathit{i}{\mathit{n}}^{-1}0\right|$
= 2a $\left(\frac{\mathit{\pi }}{2}\right)$
= πa