The value of


The value of $\int_{-\pi / 2}^{\pi / 2} \frac{d x}{[x]+[\sin x]+4}$, where $[t]$

denotes the greatest integer less than or equal to $\mathrm{t}$, is :

  1. $\frac{1}{12}(7 \pi+5)$

  2. $\frac{3}{10}(4 \pi-3)$

  3. $\frac{1}{12}(7 \pi-5)$

  4. $\frac{3}{20}(4 \pi-3)$

Correct Option: , 4


$I=\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \frac{d x}{[x]+[\sin x]+4}$

$=\int_{\frac{-\pi}{2}}^{-1} \frac{d x}{-2-1+4}+\int_{-1}^{0} \frac{d x}{-1-1+4}$

$+\int_{0}^{1} \frac{d x}{0+0+4}+\int_{1}^{\frac{\pi}{2}} \frac{d x}{1+0+4}$

$\int_{\frac{-\pi}{2}}^{-1} \frac{d x}{1}+\int_{-1}^{0} \frac{d x}{2}+\int_{0}^{1} \frac{d x}{4}+\int_{1}^{\frac{\pi}{2}} \frac{d x}{5}$



$\frac{-20+10+5-4}{20}+\frac{6 \pi}{10}$

$\frac{-9}{20}+\frac{3 \pi}{5}$

Option (4)

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