$\int\limits_{ - 1}^1 {\left( {x + 1} \right)dx}$

$\int\limits_{ - 1}^1 {\left( {x + 1} \right)dx} = \left[ {\cfrac{{{x^2}}}{2} + x} \right]_{ - 1}^1 = \cfrac{1}{2}\left[ {{{\left( 1 \right)}^2} - {{\left( { - 1} \right)}^2} + \left[ {1 - \left( { - 1} \right)} \right]} \right]$

$= \cfrac{1}{2}\left( {1 - 1} \right) + \left( {1 + 1} \right) = \cfrac{1}{2}\left( 0 \right) + 2 = 2$