Let $$f\left( x \right) = x - \left[ x \right]$$    for $$x\, \in \,R,$$   where $$\left[ x \right] = $$  the greatest integer $$ \leqslant x.$$  Then $$\int_{ - 2}^2 {f\left( x \right)dx} $$   is :

Solution :
$$\eqalign{ & I = \int_{ - 2}^2 {x\,dx} - \int_{ - 2}^2 {\left[ x \right]dx} \cr & = 0 - \left\{ {\int_{ - 2}^{ - 1} {\left[ x \right]dx + \int_{ - 1}^0 {\left[ x \right]dx} + \int_0^1 {\left[ x \right]dx} + \int_1^2 {\left[ x \right]dx} } } \right\} \cr & = - \int_{ - 2}^{ - 1} { - 2\,dx} - \int_{ - 1}^0 { - 1\,dx} - \int_0^1 {0\,dx} - \int_1^2 {1\,dx} \cr & = 2\left| x \right|_{ - 2}^{ - 1} + \left| x \right|_{ - 1}^0 - 0 - \left| x \right|_1^2 \cr & = 2\left( { - 1 + 2} \right) + \left( {0 + 1} \right) - \left( {2 - 1} \right) \cr & = 2 \cr} $$