A) \[\frac{1}{\sin (a-b)}\log \left| \frac{\sin (x-a)}{\sin (x-b)} \right|+c\]
B) \[\frac{-1}{\sin (a-b)}\log \left| \frac{\sin (x-a)}{\sin (x-b)} \right|+c\]
C) \[\log \sin (x-a)\sin (x-b)+c\]
D) \[\log \left| \frac{\sin (x-a)}{\sin (x-b)} \right|\]
E) \[\frac{1}{\sin (x-a)}\log \sin (x-a)\sin (x-b)+c\]
Correct Answer: A
Solution :
Let \[\int{\frac{dx}{\sin (x-a)\sin (x-b)}}\] \[=\frac{1}{\sin (a-b)}\int{\frac{\sin \left\{ (x-b)-(x-a) \right\}}{\sin (x-a)\sin (x-b)}}\ dx\] \[=\frac{1}{\sin (a-b)}\int{\frac{\sin (x-b)\cos (x-a)-\cos (x-b)\sin (x-a)}{\sin (x-a)\sin (x-b)}dx}\] \[=\frac{1}{\sin (a-b)}\left[ \int{\cot (x-a)dx-\int{\cot (x-b)dx}} \right]\] \[=\frac{1}{\sin (a-b)}\ \left[ \log \sin (x-a)-\log \sin (x-b) \right]+c\] \[=\frac{1}{\sin (a-b)}\ \log \left| \frac{\sin (x-a)}{\sin (x-b)} \right|+c\].
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