Evaluate (x-a) (x+a) d x - Vidyadip

Question

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

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Answer

According to the question,$I=\int \frac{\sin (x-a)}{\sin (x+a)} d x$
$\text { Put } x+a=t \Rightarrow dx=dt$
$\therefore I=\int \frac{\sin (t-a-a)}{\sin t} d t=\int \frac{\sin (t-2 a)}{\sin t} d t$
$=\int \frac{\sin t \cos 2 a-\cos t \sin 2 a}{\sin t} d t$
${(\therefore \sin (A-B)=\sin A \cos B-\cos A \sin B)}$
$=\int \cos 2 a d t-\int \sin 2 a \cdot \cot t d t$
$=\cos 2 a(t)-\sin 2 a(\log |\sin t|)+C_1$
$=(x+a) \cos 2 a-\sin 2 a \log |\sin (x+a)|+C_1$
$($put $t=x+a)$
$=x \cos 2 a-\sin 2 a \log |\sin (x+a)|+C _1$

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