A) \[x\cos \alpha -\sin \alpha \log \sin (x-\alpha )+c\]
B) \[x\cos \alpha +\sin \alpha \log \sin (x-\alpha )+c\]
C) \[x\sin \alpha -\sin \alpha \log \sin (x-\alpha )+c\]
D) None of these
Correct Answer: B
Solution :
\[\int_{{}}^{{}}{\frac{\sin x}{\sin (x-\alpha )}\,dx=\int_{{}}^{{}}{\frac{\sin (x-\alpha +\alpha )}{\sin (x-\alpha )}\,dx}}\] \[=\int_{{}}^{{}}{\frac{\left\{ (\sin (x-\alpha )\cos \alpha +\cos (x-\alpha )\sin \alpha \right\}}{\sin (x-\alpha )}\,dx}\] \[=\int_{{}}^{{}}{\cos \alpha \,dx+\int_{{}}^{{}}{\sin \alpha \,.\,\cot \,(x-\alpha )\,dx}}\] \[=x\cos \alpha +\sin \alpha \,.\,\log \sin (x-\alpha )+c\].
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