Question

The value of $I={\int }_0^3\left(\right[x]+[x+\frac{1}{3}]+[x+\frac{2}{3}])dx$, where $[\cdot ]$ denotes the greatest integer function, is equal to

• A. 10
  
• B. 11
  
• C. 12
  
• D. 14

Answer 1

The correct option is C

12

Let $I={\int }_0^{\frac{1}{3}}0dx+{\int }_{\frac{1}{3}}^{\frac{2}{3}}1dx+{\int }_{\frac{2}{3}}^{1}2dx+{\int }_1^{\frac{4}{3}}3dx+{\int }_{\frac{4}{3}}^{\frac{5}{3}}4dx+{\int }_{\frac{5}{3}}^{\frac{6}{3}}5dx+{\int }_{\frac{6}{3}}^{\frac{7}{3}}6dx+{\int }_{\frac{7}{3}}^{\frac{8}{3}}7dx+{\int }_{\frac{8}{3}}^{\frac{9}{3}}8dx=\frac{1}{3}(1+2+3+4+5+6+7+8)=12$