Question

Let $f\left(x\right)=$ maximum $\left\{x+\left|x\right|,x-\left[x\right]\right\},$ where$\left[x\right]$ is the greatest integer less then or equal to $x,$ then $\underset{-2}{\overset{2}{\int }}f\left(x\right)dx=$

• A. $1$
• B. $3$
• C. $5$
• D. $7$

Answer 1

The correct option is C $5$

$x+\left|x\right|$
$whenx>0$
$x+\left|x\right|=2x$
$whenx<0$
$x+\left|x\right|=0$
$x-\left[x\right]=\left[x\right]$
$\underset{-2}{\overset{2}{\int }}f\left(x\right)dx=areaunderthecu$
$rves$
$=areaof\text{small triangles+area of larger triangle}$
$=2[\frac{1}{2}\times 1\times 1]+\frac{1}{2}\times 2\times 4$
$=1+4=5$