• # question_answer The value of $\int\limits_{{\scriptstyle{}^{-\pi }/{}_{2}}}^{{\scriptstyle{}^{\pi }/{}_{2}}}{\frac{dx}{[x]+[sinx]+4},}$ where $[t]$ denotes the greatest integer less than or equal to $t,$ is: A) $\frac{1}{12}(7\pi +5)$ B) $\frac{1}{12}(7\pi -5)$ C) $\frac{3}{20}(4\pi -3)$ D) $\frac{3}{10}(4\pi -3)$

 $\int\limits_{-\pi /2}^{\pi /2}{\frac{dx}{[x]+[sinx]+4}}$
 $=\int\limits_{-\pi /2}^{0}{\frac{dx}{[x]+-1+4}+}\int\limits_{0}^{\pi /2}{\frac{dx}{[x]+4}}$ $=\int\limits_{-\pi /2}^{-1}{\frac{dx}{-2-1+4}+}\int\limits_{-1}^{0}{\frac{dx}{-1-1+4}}+\int\limits_{0}^{1}{\frac{dx}{4}+\int\limits_{1}^{\pi /2}{\frac{dx}{1+4}}}$ $=-1+\frac{\pi }{2}+2+\frac{1}{4}+\frac{1}{5}\left( \frac{\pi }{2}-1 \right)$ $=3\frac{\pi }{5}-\frac{9}{20}.$