Question

Solve :
$\int \frac{dx}{sinx+cosx}$

Answer 1

$\int \frac{dx}{sinx+cosx}$

$=\int \frac{dx}{\sqrt{2}[\frac{1}{\sqrt{2}}sinx+\frac{1}{\sqrt{2}}cosx]}$

$=\int \frac{dx}{\sqrt{2}[sinx\cos \frac{\pi }{4}+\sin \frac{\pi }{4}.cosx]}$

$=\int \frac{dx}{\sqrt{2}\sin (x+\frac{\pi }{4})}$

$=\frac{1}{\sqrt{2}}\int cosec(x+\frac{\pi }{4})dx$

$=\frac{1}{\sqrt{2}}log\left|\tan (\frac{x}{2}+\frac{\pi }{8})\right|+c$ $[\because \int cosecxdx=log\left|\tan \frac{x}{2}\right|+c]$