Show that

$\frac{\sin x}{\sin (x-a)}$

Let $x-a=t \Rightarrow d x=d t$

$\int \frac{\sin x}{\sin (x-a)} d x=\int \frac{\sin (t+a)}{\sin t} d t$

$=\int \frac{\sin t \cos a+\cos t \sin a}{\sin t} d t$

$=\int(\cos a+\cot t \sin a) d t$

$=t \cos a+\sin a \log |\sin t|+C_{1}$

$=(x-a) \cos a+\sin a \log |\sin (x-a)|+\mathrm{C}_{1}$

$=x \cos a+\sin a \log |\sin (x-a)|-a \cos a+\mathrm{C}_{1}$

$=\sin a \log |\sin (x-a)|+x \cos a+\mathrm{C}$