The value of $ I = \int_{0}^{3}([x] + [x + \frac{1}{3}] + [x+\frac{2}{3}])dx$

Q: The value of $\large I = \int_{0}^{3}([x] + [x + \frac{1}{3}] + [x+\frac{2}{3}])dx$

where [.] denotes the greatest integer function, is equal to;

(A) 10

(B) 11

(C) 12

(D) none of these

Ans: (C)

Sol: Let $\displaystyle I = \int_{0}^{1/3} 0.dx + \int_{1/3}^{2/3} 1 .dx + \int_{2/3}^{1} 2.dx + \int_{1}^{4/3} 3 .dx + \int_{4/3}^{5/3} 4.dx $

$\displaystyle + \int_{5/3}^{6/3} 5.dx + \int_{6/3}^{7/3} 6 .dx + \int_{7/3}^{8/3} 7 .dx + \int_{8/3}^{9/3} 8 .dx $

$\displaystyle I = \frac{1}{3}(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 ) $

= 12

Hence (C) is the correct answer.