If $$\int_{ - 2}^3 {f\left( x \right)} dx = 5$$    and $$\int_1^3 {\left\{ {2 - f\left( x \right)} \right\}dx = 6} $$      then the value of $$\int_{ - 2}^1 {f\left( x \right)} dx$$    is :

Solution :
$$\eqalign{ & \int_1^3 {\left\{ {2 - f\left( x \right)} \right\}dx = 6\,\,\,\,\,\,\,\,\,} \Rightarrow 2\left( {3 - 1} \right) - 6 = \int_1^3 {f\left( x \right)dx} \cr & \therefore \int_1^3 {f\left( x \right)dx = - 2} \cr & {\text{Now, }}\int_{ - 2}^1 {f\left( x \right)dx} = \int_{ - 2}^3 {f\left( x \right)dx} - \int_1^3 {f\left( x \right)dx} = 5 - \left( { - 2} \right) = 7 \cr} $$