Question:
$\frac{\sin x}{\sin (x-a)}$
Solution:
$\frac{\sin x}{\sin (x-a)}$
Let $x-a=t \Rightarrow d x=d t$
$\int \frac{\sin x}{\sin (x-a)} d x=\int \frac{\sin (t+a)}{\sin t} d t$
$=\int \frac{\sin t \cos a+\cos t \sin a}{\sin t} d t$
$=\int(\cos a+\cot t \sin a) d t$
$=t \cos a+\sin a \log |\sin t|+C_{1}$
$=(x-a) \cos a+\sin a \log |\sin (x-a)|+\mathrm{C}_{1}$
$=x \cos a+\sin a \log |\sin (x-a)|-a \cos a+\mathrm{C}_{1}$
$=\sin a \log |\sin (x-a)|+x \cos a+\mathrm{C}$