Question

Evaluate:
$\int \sqrt{\frac{a+x}{a-x}}dx$.

Answer 1

$\begin{array}{c}\int \sqrt{\frac{a+x}{a-x}}dx\\ x=a\cos 2\theta \\ dx=-2\sin 2\theta d\theta \\ \sqrt{\frac{a+a\cos 2\theta }{a-a\cos 2\theta }}\left(-2a\sin 2\theta d\theta \right)\\ \sqrt{\frac{a\left(1+\cos 2\theta \right)}{a\left(1-\cos 2\theta \right)}}\left(-2asin2\theta d\theta \right)\\ \sqrt{\frac{\cos 2\theta }{\sin 2\theta }}\left(-2\sin 2\theta d\theta \right)\\ \int \cot \theta \left(-2a\sin 2\theta d\theta \right)\\ =-2a\int 2\cos 2\theta d\theta \\ =-2a\int \left(1+\cos 2\theta \right)d\theta \\ =-2a\left(\theta +\frac{\sin 2\theta }{2}\right)\\ =-2a\theta -a\sin 2\theta \\ =-a{\cos }^{-1}\frac{x}{a}-\sqrt{a^2-a^2{\cos }^{2}2\theta }\\ =-a{\cos }^{-1}\frac{x}{a}-\sqrt{a^2-x^2}+c\end{array}$