Evaluate: (x - ) (x + ) dx - Vidyadip

Question

Evaluate:$\int \frac{\sin (\text{x} - \alpha)}{\sin (\text{x} + \alpha)} \text{dx}$

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Answer

$I = \int \frac{\sin(x + a)-2\alpha)}{\sin(x + \alpha)}$$= \int \frac{\sin(x + \alpha). \cos 2\alpha - \cos (x + \alpha) .2 \alpha}{\sin(x + \alpha)} dx$
$= \cos 2 \alpha \int dx - \sin 2 \alpha \int \frac{\cos(x + \alpha)}{\sin(x + \alpha)} dx$
$= x \cos 2 \alpha - \sin 2 \alpha \log |\sin (x + \alpha)|+c$

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