Question

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

• A. $(x-a)\cos a-\sin a\log \sin (x-a)$
• B. $(x-a)\cos a+\sin a\log \sin (x-a)$
• C. $(x-a)\sin a+\cos a\log \sin (x-a)$
• D. $(x-a)\sin a-\cos a\log \sin (x-a)$

Answer 1

The correct option is B $(x-a)\cos a+\sin a\log \sin (x-a)$

Let $I=\int \frac{\sin x}{\sin (x-a)}dx$
Put $x-a=t\Rightarrow dx=dt$
Therefore,$I=\int \frac{\sin (a+t)}{\sin t}dt=\int \frac{\sin a\cos t+\cos a\sin t}{\sin t}dt$
$=\int (\sin a\cot t+\cos a)dt=\sin a\log \csc t+t\cos a$
$=(x-a)\cos a+\sin a\log \sin (x-a)$.