a) \(\int\limits_0^1 {\left( {4{x^3} + x} \right)dx} = \left. {\left( {{x^4} + \frac{{{x^2}}}{2}} \right)} \right|_0^1 = \frac{3}{2} - 0 = \frac{3}{2}\)
b) \(\int\limits_1^2 {\frac{{x - 2}}{{{x^2}}}dx} = \int\limits_1^2 {\left( {\frac{1}{x} - \frac{2}{{{x^2}}}} \right)dx} = \int\limits_1^2 {\left( {\frac{1}{x} - 2{x^{ - 2}}} \right)dx} = \left. {\left( {\ln \left| x \right| - 2\frac{{{x^{ - 1}}}}{{ - 1}}} \right)} \right|_1^2\)
\( = \left. {\left( {\ln \left| x \right| + \frac{2}{x}} \right)} \right|_1^2 = \left( {\ln 2 + 1} \right) - \left( {\ln 1 + 2} \right) = \ln 2 - 1\)
c) \(\int\limits_0^4 {{2^{2x}}dx} = \int\limits_0^4 {{4^x}dx} = \left. {\left( {\frac{{{4^x}}}{{\ln 4}}} \right)} \right|_0^4 = \frac{{{4^4}}}{{\ln 4}} - \frac{{{4^0}}}{{\ln 4}} = \frac{{255}}{{\ln 4}}\)
d) \(\int\limits_1^2 {\left( {{e^{x - 1}} + {2^{x + 1}}} \right)dx} = \int\limits_1^2 {\left( {\frac{{{e^x}}}{e} + {2^x}.2} \right)dx} = \frac{1}{e}\int\limits_1^2 {{e^x}dx} + 2\int\limits_1^2 {{2^x}dx} \)
\( = \frac{1}{e}.\left. {\left( {{e^x}} \right)} \right|_1^2 + 2.\left. {\left( {\frac{{{2^x}}}{{\ln 2}}} \right)} \right|_1^2 = \frac{1}{e}\left( {{e^2} - {e^1}} \right) + 2.\left( {\frac{{{2^2}}}{{\ln 2}} - \frac{{{2^1}}}{{\ln 2}}} \right) = e - 1 + 2.\frac{3}{{\ln 2}} = e - 1 + \frac{6}{{\ln 2}}\)