Question

The value of $I=\sqrt{2}\int \frac{\sin x}{\sin \left(x-\frac{\pi }{4}\right)}dx$ is

Answer 1

$x-\frac{\pi }{4}=t$

$\Rightarrow x=t+\frac{\pi }{4}$

$dx=dt$

$\therefore I=\sqrt{2}\int \frac{\sin (\pi /4+t)}{\sin \left(t\right)}dt$

$\Rightarrow \int \sqrt{2}\frac{(\frac{1}{\sqrt{2}}\cdot \sin t+\frac{1}{\sqrt{2}}\cos t)}{\sin t}dt$

$\Rightarrow \int (\frac{\sin t}{\sin t}+\frac{\cos t}{\sin t})dt$

$\Rightarrow \int (1+\cot t)dt=t+logcosect+c$

$\Rightarrow I=(x-\pi /4)+logcosec(-\pi /4+x)+c$.