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