A) 2
B) \[\frac{3}{4}\]
C) 0
D) None of these
Correct Answer: C
Solution :
Let \[I=\int_{0}^{\pi /2}{\log \left( \frac{4+3\sin x}{4+3\cos x} \right)}\,dx.\] Then, \[I=\int_{0}^{\pi /2}{\log \left( \frac{4+3\cos x}{4+3\sin x} \right)}\,dx\], \[\left[ \because \int_{0}^{\pi /2}{f(x)dx=\int_{0}^{\pi /2}{f\left( \frac{\pi }{2}-x \right)\,dx}} \right]\] Þ \[I=-\int_{0}^{\pi /2}{\log \left( \frac{4+3\sin x}{4+3\cos x} \right)\,dx=-I}\] Þ \[2I=0\Rightarrow I=0\].
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