Question

Integrate the following function $\sqrt{\frac{1+\sin x}{1-\sin x}}$

Answer 1

${\int }^{}\sqrt{\frac{1+\sin x}{1-\sin x}}dx$

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

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

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

$={\int }^{}\frac{1+\sin x}{\cos x}dx$

$={\int }^{}\left(\sec x+\tan x\right)dx$

$=\log \left|\sec x+\tan x\right|+\log \left|\sec x\right|+c$

$=\log \left|\left(\sec x+\tan x\right)\sec x\right|+c$

$=\log \left|{\sec }^{2}x+\sec x\tan x\right|+c$

$=\log \left|\frac{1+\sin x}{{\cos }^{2}x}\right|+c$

$=\log \left|1+\sin x\right|-2\log \left|\cos x\right|+c$