Question

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

Answer 1

Consider the given function:

$I=\int \sqrt{\frac{a+x}{a+x}}dx$

$=\int \sqrt{\frac{(a+x)(a+x)}{(a+x)(a+x)}}dx$

$=\int \frac{(a+x)}{\sqrt{a^2-x^2}}dx$

$=\int \frac{a}{\sqrt{a^2-x^2}}dx+\int \frac{x}{\sqrt{a^2-x^2}}dx$

$=a{\sin }^{-1}\frac{x}{a}+\left(\frac{1}{2}\right)\int \frac{-2x}{\sqrt{a^2-x^2}}dx$

$let{a}^2-x^2=t^2$

$-2xdx=2tdx$

$=a{\sin }^{-1}\frac{x}{a}-\frac{1}{2}\int \frac{2t}{t}dx$

$=a{\sin }^{-1}\frac{x}{2}-\frac{1}{2}\int 1dt$

$=a{\sin }^{-1}\frac{x}{2}-t$

$=a{\sin }^{-1}\frac{x}{2}-\sqrt{a^2-x^2}+c$

Hence this is the answer.