a) \(\int\limits_0^1 {\left( {4{x^3} + x} \right)dx} \)\( = 4\int\limits_0^1 {{x^3}dx} + \int\limits_0^1 {xdx} \)\( = \left. {\left( {{x^4} + \frac{{{x^2}}}{2}} \right)} \right|_0^1 = \frac{3}{2}\).

b) \(\int\limits_1^2 {\frac{{x - 2}}{{{x^2}}}dx} \)\( = \int\limits_1^2 {\frac{1}{x}dx} - 2\int\limits_1^2 {\frac{1}{{{x^2}}}dx} \)\( = \left. {\left( {\ln \left| x \right| + \frac{2}{x}} \right)} \right|_1^2\)= ln2 + 1 – 2 = ln2 – 1.

c) \(\int\limits_0^4 {{2^{2x}}dx} \)\( = \int\limits_0^4 {{4^x}dx} \)\( = \left. {\left( {\frac{{{4^x}}}{{\ln 4}}} \right)} \right|_0^4 = \frac{{256}}{{\ln 4}} - \frac{1}{{\ln 4}} = \frac{{255}}{{\ln 4}}\).

d) \(\int\limits_1^2 {\left( {{e^{x - 1}} + {2^{x + 1}}} \right)dx} \)\( = \frac{1}{e}\int\limits_1^2 {{e^x}dx} + 2\int\limits_1^2 {{2^x}dx} \)\( = \left. {\left( {\frac{1}{e}.{e^x} + 2.\frac{{{2^x}}}{{\ln 2}}} \right)} \right|_1^2\)\( = e + \frac{4}{{\ln 2}} - 1\).