Question

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

• A. $2$
• B. $3$
• C. $-3$
• D. none of these

Answer 1

Answer: (B)

$3$

$I={\int }_0^2[x+[x+\left[x\right]]]dx={\int }_0^2[x+2\left[x\right]]dx$
$\because [x+n]=\left[x\right]+n\Rightarrow [x+\left[x\right]]=\left[x\right]+\left[x\right]\therefore I={\int }_0^2[x+2\left[x\right]]dx={\int }_0^2\left[3\left[x\right]\right]dx=3\left\{{\int }_0^1\left[x\right]dx+{\int }_1^2\left[x\right]dx\right\}=3\left\{{\int }_0^{1}0dx+{\int }_1^2\left[x\right]dx\right\}=3(2-1)=3$