Question

Integrate $\int \sqrt{\frac{a-x}{a+x}}dx$

Answer 1

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

substitute $x=a\cos \theta \to dx=-a\sin \theta d\theta$

$-\int \sqrt{\frac{a-a\cos \theta }{a+a\cos \theta }}-a\sin \theta d\theta$

$-\int \sqrt{\frac{a\left(2{\sin }^2\frac{\theta }{2}\right)}{a\left(2{\cos }^2\frac{\theta }{2}\right)}}-a2\sin \frac{\theta }{2}\cos \frac{\theta }{2}d\theta$

$-2a\int {\sin }^2\frac{\theta }{2}d\theta$

$-2a\int \frac{1}{2}(1-\cos \theta )d\theta$

$-a[\theta -\sin \theta ]+C$

$-a[{\cos }^{-1}\left(\frac{x}{a}\right)-\sqrt{1-{\left(\frac{x}{a}\right)}^2}]+C$