a) \(\int\limits_0^{2\pi } {\left( {2x + \cos x} \right)dx} = 2\int\limits_0^{2\pi } {xdx} + \int\limits_0^{2\pi } {\cos xdx} = 2.\frac{{{x^2}}}{2}\left| \begin{array}{l}2\pi \\0\end{array} \right. + \sin x\left| \begin{array}{l}2\pi \\0\end{array} \right.\)
\( = {\left( {2\pi } \right)^2} - 0 + \sin 2\pi - \sin 0 = 4{\pi ^2}\)
b) \(\int\limits_1^2 {\left( {{3^x} - \frac{3}{x}} \right)dx} = \int\limits_1^2 {{3^x}dx} - 3\int\limits_1^2 {\frac{1}{x}dx} = \frac{{{3^x}}}{{\ln 3}}\left| \begin{array}{l}2\\1\end{array} \right. - 3\ln \left| x \right|\left| \begin{array}{l}2\\1\end{array} \right. = \frac{1}{{\ln 3}}\left( {{3^2} - {3^1}} \right) - 3\ln 2 + 3\ln 1\)
\( = \frac{6}{{\ln 3}} - 3\ln 2\)
c) \(\int\limits_{\frac{\pi }{6}}^{\frac{\pi }{3}} {\left( {\frac{1}{{{{\cos }^2}x}} - \frac{1}{{{{\sin }^2}x}}} \right)dx} = \int\limits_{\frac{\pi }{6}}^{\frac{\pi }{3}} {\frac{1}{{{{\cos }^2}x}}dx} - \int\limits_{\frac{\pi }{6}}^{\frac{\pi }{3}} {\frac{1}{{{{\sin }^2}x}}dx = \tan x\left| \begin{array}{l}\frac{\pi }{3}\\\frac{\pi }{6}\end{array} \right. + \cot x\left| \begin{array}{l}\frac{\pi }{3}\\\frac{\pi }{6}\end{array} \right.} \)
\( = \tan \frac{\pi }{3} - \tan \frac{\pi }{6} + \cot \frac{\pi }{3} - \cot \frac{\pi }{6} = \sqrt 3 - \frac{{\sqrt 3 }}{3} + \frac{{\sqrt 3 }}{3} - \sqrt 3 = 0\)