Question

The value of ${\int }_{-1}^3(|x-2|+[x\left]\right)dx$, where $\left[x\right]$ denotes the greatest integer less than or equal to $x,$ is

• A. 7
• B. 5
• C. 4
• D. 3

Answer 1

The correct option is A

7

$|x-2|=\{\begin{array}{c}-(x-2)\text{for}-1\le x<2\\ (x-2)\text{for}2\le x\le 3\end{array}$

$\left[x\right]=\{\begin{array}{c}-1\text{for}-1\le x<0\\ 0\text{for}0\le x<1\\ 1\text{for}1\le x<2\\ 2\text{for}2\le x<3\end{array}$

$\text{So,}{\int }_{-1}^3(|x-2|+[x\left]\right)dx={\int }_{-1}^2-(x-2)dx+{\int }_2^3(x-2)dx+{\int }_{-1}^0(-1)dx+{\int }_0^{1}0dx+{\int }_1^{2}1dx+{\int }_2^{3}2dx.$

$={[2x-\frac{x^2}{2}]}_{-1}^2+{[\frac{x^2}{2}-2x]}_2^3+[-x{]}_{-1}^0+[x{]}_1^2+2[x{]}_2^3$

$=\frac{9}{2}+\frac{1}{2}-1+1+2=7$