Question

$\int \frac{1}{\sin x+\cos x}dx=\frac{1}{\sqrt{\left(2\right)}}\log \tan (\frac{x}{2}+\frac{\pi }{k}).$ Find the value of $k$.

Answer 1

Let $I=\int \frac{1}{\sin x+\cos x}dx$
$=\int \frac{1}{\sqrt{2}(\frac{1}{\sqrt{2}}\sin x+\frac{1}{\sqrt{\left(2\right)}}\cos x)}dx.$
$=\frac{1}{\sqrt{\left(2\right)}}\int \frac{1}{\sin (x+\pi /4)}dx$
$=\frac{1}{\sqrt{\left(2\right)}}\int \frac{1}{\sin (x+\pi /4)}dx$
$=\frac{1}{\sqrt{\left(2\right)}}\int co\sec (x+\frac{\pi }{4})dx=\frac{1}{\sqrt{\left(2\right)}}\log \tan (\frac{x}{2}+\frac{\pi }{8})$