A) \[\frac{1}{12}(7\pi +5)\]
B) \[\frac{1}{12}(7\pi -5)\]
C) \[\frac{3}{20}(4\pi -3)\]
D) \[\frac{3}{10}(4\pi -3)\]
Correct Answer: C
Solution :
| \[\int\limits_{-\pi /2}^{\pi /2}{\frac{dx}{[x]+[sinx]+4}}\] |
| \[=\int\limits_{-\pi /2}^{0}{\frac{dx}{[x]+-1+4}+}\int\limits_{0}^{\pi /2}{\frac{dx}{[x]+4}}\] |
| \[=\int\limits_{-\pi /2}^{-1}{\frac{dx}{-2-1+4}+}\int\limits_{-1}^{0}{\frac{dx}{-1-1+4}}+\int\limits_{0}^{1}{\frac{dx}{4}+\int\limits_{1}^{\pi /2}{\frac{dx}{1+4}}}\] |
| \[=-1+\frac{\pi }{2}+2+\frac{1}{4}+\frac{1}{5}\left( \frac{\pi }{2}-1 \right)\] |
| \[=3\frac{\pi }{5}-\frac{9}{20}.\] |
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