Question

If $\left[x\right]$ denotes the greater integer less than or equal to x, then the value of ${\int }_1^5[|x-3|]$dx is

• A. 1
• B. 2
• C. 4
• D. 8

Answer 1

The correct option is B 2

$I{\int }_1^5[|x-3|]dx\Rightarrow I={\int }_1^3[-(x-3\left)\right]dx+{\int }_3^5\left[\right(x-3\left)\right]dx\Rightarrow {\int }_1^2[-(x-3\left)\right]dx+{}_2^3[-(x-3\left)\right]dx+{}_3^4[x-3]dx+{}_4^5[x-3]dx\Rightarrow I={\int }_1^{2}dx+{\int }_2^{3}0dx+{\int }_3^{4}0dx+{\int }_4^{5}dx={\left[x\right]}_1^2+{\left[x\right]}_4^5\Rightarrow I=(2-1)+(5-4)=2.$