Biết rằng \[\int\limits_0^{\frac{\pi }{3}} {{{\sin }^2}x\cos xdx} = \frac,\] với \[a,{\rm{ }}b \in \mathbb{Z}.\] Tính \[S = a + 2b.\]

Chọn đáp án A

Ta có: \(\int\limits_0^{\frac{\pi }{3}} {{{\sin }^2}x\cos xdx} = \int\limits_0^{\frac{\pi }{3}} {{{\sin }^2}xd\left( {\sin x} \right)} = \left. {\frac{{{{\sin }^3}x}}{3}} \right|_0^{\frac{\pi }{3}} = \frac{{2\sqrt 3 }} \Rightarrow \left\{ \begin{array}{l}a = 0\\b = 2\end{array} \right. \Rightarrow S = 4\).