Question

Let $f\left(x\right)=max(x+\left|x\right|,x-\left[x\right])$ where [x] = the greatest integer in $x\le x$. Then ${\int }_{-2}^{2}f\left(x\right)dx$ is equal to

• A. 3
• B. 2
• C. 1
• D. None of these

Answer 1

The correct option is D

None of these

${\int }_{-2}^{2}f\left(x\right)dx={\int }_{-2}^0(x-[x\left]\right)dx+{\int }_0^2(x+\left|x\right|)dx={\int }_{-2}^{0}xdx-{\int }_{-2}^0\left[x\right]dx+{\int }_0^{2}2xdx={\left|\frac{x^2}{2}\right|}_{-2}^0-\left[{\int }_{-2}^{-1}\right[x]dx+{\int }_{-1}^0[x\left]dx\right]+{\left|x^2\right|}_0^2=0-2-[{\int }_{-2}^{-1}-2dx+{\int }_{-1}^0-1dx]+4=2+{\left|2x\right|}_{-2}^{-1}+{\left|x\right|}_{-1}^0=2+[-2+4]+[0+1]=5$