Question

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

Answer 1

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

$={\int }^{}\frac{1}{\sqrt{{\sin }^2\frac{x}{2}+{\cos }^2\frac{x}{2}-2\sin \frac{x}{2}\cos \frac{x}{2}}}dx$

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

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

$=\frac{1}{\sqrt{2}}{\int }^{}\frac{dx}{\sin \frac{x}{2}\cos \frac{\pi }{4}-\cos \frac{x}{2}\sin \frac{\pi }{4}}$

$=\frac{1}{\sqrt{2}}{\int }^{}\frac{dx}{\sin \left(\frac{x}{2}-\frac{\pi }{4}\right)}$

$=\frac{1}{\sqrt{2}}{\int }^{}\cos ec\left(\frac{x}{2}-\frac{\pi }{4}\right)dx$

$=\frac{1}{\sqrt{2}}\log \left|\cos ec\left(\frac{x}{2}-\frac{\pi }{4}\right)-\cot \left(\frac{x}{2}-\frac{\pi }{4}\right)\right|+c$

$=\frac{1}{\sqrt{2}}\log \left|\frac{1-\cos \left(\frac{x}{2}-\frac{\pi }{4}\right)}{\sin \left(\frac{x}{2}-\frac{\pi }{4}\right)}\right|+c$

$=\frac{1}{\sqrt{2}}\log \left|\frac{2{\sin }^2\left(\frac{x}{4}-\frac{\pi }{8}\right)}{2\sin \left(\frac{x}{4}-\frac{\pi }{8}\right)\cos \left(\frac{x}{4}-\frac{\pi }{8}\right)}\right|+c$

$=\frac{1}{\sqrt{2}}\log \left|\tan \left(\frac{x}{4}-\frac{\pi }{8}\right)\right|+c$