The value of ∫ _-π /2^π /2 dx [x]+[ sin⁡x] + 4, where [t] denotes the greatest integer less than or equal to t, is

We have$\displaystyle {\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\frac{dx}{[x]+[sinx]+4}$∫−2π​2π​​[x]+[sinx]+4dx​

$$={\int }_{-\frac{\pi }{2}}^{0}\frac{dx}{[x]+3}+{\int }_{0}^{\frac{\pi }{2}}\frac{dx}{[x]+4}$$=∫−2π​0​[x]+3dx​+∫02π​​[x]+4dx​

$$={\int }_{-\frac{\pi }{2}}^{-1}\frac{dx}{1}+{\int }_{-1}^{0}\frac{dx}{2}+{\int }_{0}^{1}\frac{dx}{4}+{\int }_{1}^{\frac{\pi }{2}}\frac{dx}{5}$$=∫−2π​−1​1dx​+∫−10​2dx​+∫01​4dx​+∫12π​​5dx​

$$={[x]}_{-\frac{\pi }{2}}^{-1}+{[\frac{x}{2}]}_{-1}^{0}+{[\frac{x}{4}]}_{0}^{1}+{[\frac{x}{5}]}_{1}^{\frac{\pi }{2}}=(-1+\frac{\pi }{2})+(0+\frac{1}{2})+\frac{1}{4}+\frac{\pi }{10}-\frac{1}{5}$$=[x]−2π​−1​+[2x​]−10​+[4x​]01​+[5x​]12π​​=(−1+2π​)+(0+21​)+41​+10π​−51​

$$=\frac{-20+10\pi +10+5+2\pi -4}{20}=\frac{12\pi -9}{20}=\frac{3}{20}(4\pi -3)$$=20−20+10π+10+5+2π−4​=2012π−9​=203​(4π−3)