$\begin{aligned} & \text { Let } I=\int\left(\sqrt{\frac{a+x}{a-x}}+\sqrt{\frac{a-x}{a+x}}\right) d x \\ & \text { Put } x=a \cos 2 \theta \Rightarrow d x=-2 a \sin 2 \theta d \theta \\ & \therefore I=-\int\left(\sqrt{\frac{a+a \cos 2 \theta}{a-a \cos 2 \theta}}+\sqrt{\frac{a-a \cos 2 \theta}{a+a \cos 2 \theta}}\right) \\ & \qquad \times 2 a \sin 2 \theta d \theta\end{aligned}$
$\begin{aligned} & =-\int\left(\sqrt{\frac{2 \cos ^2 \theta}{2 \sin ^2 \theta}}+\sqrt{\frac{2 \sin ^2 \theta}{2 \cos ^2 \theta}}\right) 2 a \sin 2 \theta d \theta \\ & =-2 a \int\left(\frac{\cos \theta}{\sin \theta}+\frac{\sin \theta}{\cos \theta}\right) \sin 2 \theta d \theta \\ & =-2 a \int \frac{1}{\sin \theta \cos \theta} \sin 2 \theta d \theta \\ & =-4 a \int d \theta=-4 a \theta+C_1 \\ & =-2 a \cos ^{-1}\left(\frac{x}{a}\right)+C_1 \\ & =-2 a\left[\frac{\pi}{2}-\sin ^{-1}\left(\frac{x}{a}\right)\right]+C_1 \\ & =2 a \sin ^{-1}\left(\frac{x}{a}\right)+C_1-\pi a \\ & =2 a \sin ^{-1}\left(\frac{x}{a}\right)+C, \text { where } C=C_1-\pi a\end{aligned}$