A) \[x+\log \left| \sin \left( x-\frac{\pi }{4} \right) \right|+c\]
B) \[x-\log \left| \cos \left( x-\frac{\pi }{4} \right) \right|+c\]
C) \[x+\log \left| \cos \left( x-\frac{\pi }{4} \right) \right|+c\]
D) \[x-\log \left| \sin \left( x-\frac{\pi }{4} \right) \right|+c\]
Correct Answer: A
Solution :
Let \[x-\frac{\pi }{4}=t\Rightarrow x=\frac{\pi }{4}+t\] \[dx=dt\] \[\therefore \,\sqrt{2}\int{\frac{\sin x}{\sin \left( x-\frac{\pi }{4} \right)}dx=\sqrt{2}\int{\frac{\sin \left( t-\frac{\pi }{4} \right)}{\sin t}}}\] \[\int{\left( 1+\cot t \right)dt=t+{{\log }_{e}}\left| \sin t \right|+c}\] \[=x-\frac{\pi }{4}+{{\log }_{e}}\left| \sin \left( x-\frac{\pi }{4} \right) \right|+c=x+{{\log }_{e}}\]\[\left| \sin \left( x-\frac{\pi }{4} \right) \right|+c\]
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