Question

Evaluate : $\int \frac{\sin (x-a)}{\sin (x+a)}dx$.

Answer 1

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

Put $x+a=t$, we have $dx=dt$
$I=\int \frac{\sin (t-2a)}{\sin t}dt$

$I=\int (\frac{\sin t\cos 2a}{\sin t}-\frac{\cos t\sin 2a}{\sin t})dt$

$I=\cos 2a\int dt-\sin 2a\int \cot tdt$

$I=t\cos 2a-\sin 2a.\log \left|\sin t\right|+C$

$I=(x+a)\cos 2a-\sin 2a.\log \left|\sin \right(x+a\left)\right|+C$