Question

The value of ${\int }_0^{\pi /4}\frac{1+\tan x}{1-\tan x}dx$ is

Answer 1

$\begin{array}{c}{\int }_0^{\frac{\pi }{4}}\frac{\left(1+\tan x\right)}{\left(1-\tan x\right)}dx\\ \Rightarrow {\int }_0^{\frac{\pi }{4}}\tan \left(\frac{\pi }{4}+x\right)dx\\ \Rightarrow -{\left[\log \left|\cos \left(\frac{\pi }{4}+x\right)\right|\right]}_0^{\frac{\pi }{4}}+c\\ \Rightarrow -\log \left|\cos \frac{\pi }{2}-\cos \frac{\pi }{4}\right|+c\\ \Rightarrow -\log \left|-\frac{1}{\sqrt{2}}\right|+c\\ \Rightarrow -\log 2^{\frac{-1}{2}}+c\\ \frac{1}{2}\log 2+c\end{array}$