∫limits_-1^1[x]+|x| d x has the value is :

$\int\limits_{-1}^1([x]+|x|) d x=\int\limits_{-1}^0(-1-x) d x+\int\limits_0^1(0+x) d x$

$=\left|-x-\frac{x^2}{2}\right|_{-1}^0+\left|\frac{x^2}{2}\right|_0^1=0-\left(1-\frac{1}{2}\right)+\left(\frac{1}{2}-0\right)=0$

Hence (1) is the correct answer.