a) \(\int\limits_0^{\frac{\pi }{4}} {(3\sin x - 2\cos x)dx} = 3\int\limits_0^{\frac{\pi }{4}} {\sin xdx} - 2\int\limits_0^{\frac{\pi }{4}} {\cos xdx} \)
\( = - 3\cos x\left| {\begin{array}{*{20}{c}}{^{\frac{\pi }{4}}}\\{_0}\end{array}} \right. - 2\sin x\left| {\begin{array}{*{20}{c}}{^{\frac{\pi }{4}}}\\{_0}\end{array}} \right. = - 3\left( {\cos \frac{\pi }{4} - \cos 0} \right) - 2\left( {\sin \frac{\pi }{4} - \sin 0} \right)\)
\( = - 3\left( {\frac{{\sqrt 2 }}{2} - 1} \right) - 2\left( {\frac{{\sqrt 2 }}{2} - 0} \right) = 3 - \frac{{5\sqrt 2 }}{2}\).
b) \(\int\limits_{\frac{\pi }{6}}^{\frac{\pi }{3}} {\left( {\frac{2}{{\sin x}} - \frac{3}{{{{\cos }^2}x}}} \right)dx} = 2\int\limits_{\frac{\pi }{6}}^{\frac{\pi }{3}} {\frac{1}{{\sin x}}dx} - 3\int\limits_{\frac{\pi }{6}}^{\frac{\pi }{3}} {\frac{1}{{{{\cos }^2}x}}dx} \)
\( = - 2\cot x\left| {\begin{array}{*{20}{c}}{^{\frac{\pi }{3}}}\\{_{\frac{\pi }{6}}}\end{array}} \right. - 3\tan x\left| {\begin{array}{*{20}{c}}{^{\frac{\pi }{3}}}\\{_{\frac{\pi }{6}}}\end{array}} \right. = - 2\left( {\cot \frac{\pi }{3} - \cot \frac{\pi }{6}} \right) - 3\left( {\tan \frac{\pi }{3} - \tan \frac{\pi }{6}} \right)\)
\( = - 2\left( {\frac{{\sqrt 3 }}{3} - \sqrt 3 } \right) - 3\left( {\sqrt 3 - \frac{{\sqrt 3 }}{3}} \right) = - \frac{{2\sqrt 3 }}{3}\).