A) 0
B) sin 4
C) 4
D) \[4-sin\text{ }4\]
Correct Answer: A
Solution :
Let \[I=\int\limits_{-2}^{2}{\frac{{{\sin }^{2}}x}{\left[ \frac{x}{\pi } \right]+\frac{1}{2}}}dx\] \[\Rightarrow \]\[I=\int\limits_{0}^{2}{\left( \frac{{{\sin }^{2}}x}{\left[ \frac{x}{\pi } \right]+\frac{1}{2}}+\frac{{{\sin }^{2}}(-x)}{\left[ -\frac{x}{\pi } \right]+\frac{1}{2}} \right)}dx\] \[\left( \left[ \frac{x}{\pi } \right]+\left[ -\frac{x}{\pi } \right]=-1\,as\,x\ne n\pi \right)\] \[\Rightarrow \]\[I=\int\limits_{0}^{2}{\left( \frac{{{\sin }^{2}}x}{\left[ \frac{x}{\pi } \right]+\frac{1}{2}}+\frac{{{\sin }^{2}}x}{-1-\left[ \frac{x}{\pi } \right]+\frac{1}{2}} \right)}dx=0\]
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