Question

${\int }_{-2}^{10}sgn(x-[x\left]\right)dx$ equals, where $\left[\right]$ denotes greatest integer function

• A. $-12$
• B. $10$
• C. $12$
• D. $8$

Answer 1

The correct option is D $8$

Using the definition of Signum function , $sgn\left(x\right)=1$ for $x>1,sgn\left(x\right)=-1$ for $x<1$ and $sgn\left(x\right)=0$ for $x=0$
${\int }_{-2}^{10}sgn(x-[x\left]\right)dx={\int }_{-2}^0(-1)dx+{\int }_0^{10}\left(1\right)dx=-(0+2)+(10-0)=-2+10=8$