Question

Find $\int \frac{1+{\cos }^{2}x}{1-\cos 2x}dx$.

Answer 1

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

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

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

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

$=\int (\text{cosec}{}^{2}x-\frac{1}{2})dx$

$=-\cot x-\frac{x}{2}+C$.