Question

Integrate the following: $\frac{\sin x}{\sin (x-a)}$

Answer 1

Let $I=\int \frac{\sin x}{\sin (x-a)}dx$

$=\int \sin x\csc (x-a)dx$

Take $x-a=t⟹x=a+t,dx=dt$

$\therefore I=\int \sin (a+t)\csc t$

$=\int \frac{\sin (a+t)}{\sin t}dt$

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

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

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

Substitute $\sin t=p⟹\cos tdt=dp$ in first integral

$\therefore I=\sin a\int \frac{1}{p}dp+\cos a\int dt$

$=\sin a\ln p+t\cos \left(a\right)$

Hence, $I=\sin a\ln \left(\sin \right(x-a\left)\right)+\cos \left(a\right)(x-a)$