a) \(\int\limits_{ - 2}^1 {\left| {2x + 2} \right|dx} \)\( = \int\limits_{ - 2}^{ - 1} {\left| {2x + 2} \right|dx} + \int\limits_{ - 1}^1 {\left| {2x + 2} \right|dx} \)\( = - 2\int\limits_{ - 2}^{ - 1} {\left( {x + 1} \right)dx} + 2\int\limits_{ - 1}^1 {\left( {x + 1} \right)dx} \)

\( = \left. { - \left( {{x^2} + 2x} \right)} \right|_{ - 2}^{ - 1} + \left. {\left( {{x^2} + 2x} \right)} \right|_{ - 1}^1\)\( = 1 + 3 + 1 = 5\).

b) \(\int\limits_0^4 {\left| {{x^2} - 4} \right|dx} \)\( = \int\limits_0^2 {\left| {{x^2} - 4} \right|dx} + \int\limits_2^4 {\left| {{x^2} - 4} \right|dx} \)\( = \int\limits_0^2 {\left( {4 - {x^2}} \right)dx} + \int\limits_2^4 {\left( {{x^2} - 4} \right)dx} \)

\( = \left. {\left( {4x - \frac{{{x^3}}}{3}} \right)} \right|_0^2 + \left. {\left( {\frac{{{x^3}}}{3} - 4x} \right)} \right|_2^4\)\( = \frac{{16}}{3} + \frac{{16}}{3} + \frac{{16}}{3} = 16\).

c) \(\int\limits_{ - \frac{\pi }{2}}^{\frac{\pi }{2}} {\left| {\sin x} \right|dx} \)\( = \int\limits_{ - \frac{\pi }{2}}^0 {\left| {\sin x} \right|dx} + \int\limits_0^{\frac{\pi }{2}} {\left| {\sin x} \right|dx} \)\( = - \int\limits_{ - \frac{\pi }{2}}^0 {\sin xdx} + \int\limits_0^{\frac{\pi }{2}} {\sin xdx} \)

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