A) \[-\log 2\]
B) \[\log 2\]
C) \[\frac{\pi }{2}\log 2\]
D) \[-\frac{\pi }{2}\log 2\]
Correct Answer: A
Solution :
Put \[\frac{\pi }{2}x=\theta \Rightarrow dx=\frac{2}{\pi }d\theta \]; As \[x=0\] to \[1,\] \[\theta =0\] to \[\frac{\pi }{2}\] Then it reduces to \[\frac{2}{\pi }\int_{0}^{\pi /2}{\,\,\log \sin \theta \,d\theta =\frac{2}{\pi }\left[ -\frac{\pi }{2}\log 2 \right]}\] \[=-\log 2\].
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