Question

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

Answer 1

$\begin{array}{c}\int \frac{1+{\cos }^{2}x}{1+\cos 2x}dx\\ =\int \frac{1+{\cos }^{2}x}{2{\cos }^{2}x}dx\\ =\frac{1}{2}\int \left(\frac{1}{{\cos }^{2}x}+\frac{{\cos }^{2}x}{{\cos }^{2}x}\right)\\ =\frac{1}{2}\left({\sec }^{2}x+1\right)dx\\ =\frac{1}{2}\left[\tan x+x\right]+c\end{array}$