Question

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

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

Answer 1

The correct option is D $12$

By Definition
Sgn $(x-[x\left]\right)=\{\begin{array}{c}1,\forall x>0\\ 0,ifx=0\\ -1,\forall x<0\end{array}$
$\therefore {\int }_{-2}^{10}sgn(x-[x\left]\right)dx=-{\int }_{-1}^{0}1dx+{\int }_0^{10}1dx$
$=-(x{)}_{-2}^0+(x{)}_0^{10}=2+10=12$