Question

${\int }_1^2\left[3x\right]dx$ where $[\cdot ]$ represents greatest integer function, is

• A. $4$
• B. $3$
• C. $8$
• D. $5$

Answer 1

The correct option is A $4$

$I={\int }_1^2\left[3x\right]dx$
Substitute $3x=t\Rightarrow dx=\frac{1}{3}dt$
$I=\frac{1}{3}{\int }_3^6\left[t\right]dt=\frac{1}{3}({\int }_3^4\left[t\right]dt+{\int }_4^5\left[t\right]dt+{\int }_5^6\left[t\right]dt)=\frac{1}{3}(3{\int }_3^{4}dt+4{\int }_4^{5}dt+5{\int }_5^{6}dt)=\frac{1}{3}(3{\left[t\right]}_3^4+4{\left[t\right]}_4^5+5{\left[t\right]}_5^6)=\frac{1}{3}(3+4+5)=4$