Question

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

• A. 5
• B. 6
• C. 7
• D. none of these

Answer 1

The correct option is C 7

${\int }_{-1}^3\{|x-2|+\left[x\right]\}dx$

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

$={\int }_{-1}^0\{2-x-1\}dx+{\int }_0^1\{2-x+0\}dx+{\int }_1^2\{2-x+1\}dx+{\int }_2^3\{x-2+2\}dx$

$={(x-\frac{x^2}{2})}_{-1}^0+{(2x-\frac{x^2}{2})}_0^1+{(3x-\frac{x^2}{2})}_1^2+{\left(\frac{x^2}{2}\right)}_2^3$
$=\frac{3}{2}+\frac{3}{2}+4-\frac{5}{2}+\frac{9}{2}-2=7$