Find the value of ∫09 [x+2]dx where [.] is the greatest integer function

# Find the value of ${\int }_{0}^{9} \left[\sqrt{\mathrm{x}}+2\right]\mathrm{dx}$ where [.] is the greatest integer function

1. A

31

2. B

23

3. C

22

4. D

27

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$\begin{array}{l}{\int }_{0}^{9} \left[\sqrt{\mathrm{x}}+2\right]\mathrm{dx}\\ ={\int }_{0}^{1} \left[\sqrt{\mathrm{x}}+2\right]\mathrm{dx}+{\int }_{1}^{4} \left[\sqrt{\mathrm{x}}+2\right]\mathrm{dx}+{\int }_{4}^{9} \left[\sqrt{\mathrm{x}}+2\right]\mathrm{dx}\\ ={\int }_{0}^{1} 2\mathrm{dx}+{\int }_{1}^{4} \left(1+2\right)\mathrm{dx}+{\int }_{4}^{9} \left(2+2\right)\mathrm{dx}\\ =2+9+20=31\end{array}$