Question

${\int }_0^{\pi /2}\frac{1}{\sin x+\cos x}dx$

Answer 1

$\begin{array}{c}{\int }_0^{\frac{\pi }{2}}\frac{1}{\left(\sin x+\cos x\right)}dx\\ {\int }_0^{\frac{\pi }{2}}\frac{1}{\sqrt{2}\sin \left(x+\frac{\pi }{4}\right)}dx={\int }_0^{\frac{\pi }{2}}\frac{1}{\sqrt{2}}\cos ec\left(x+\frac{\pi }{4}\right)\\ \Rightarrow \frac{1}{2}\left[\log \left|\cos ec\left(x+\frac{\pi }{4}\right)-\cot \left(x+\frac{\pi }{4}\right)\right|\right]+c\\ \Rightarrow \frac{1}{2}{\left[\log \left|\tan \left(\frac{x}{2}+\frac{\pi }{8}\right)\right|\right]}_0^{\frac{\pi }{2}}=\frac{1}{2}\left[\log \frac{0.24}{0.41}\right].\end{array}$