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Câu hỏi: Biết rằng $\int\limits_{0}^{\dfrac{\pi }{3}}{{{\sin }^{2}}x\cos xdx}=\dfrac{a+b\sqrt{3}}{16},$ với $a,b\in \mathbb{Z}.$ Tính $S=a+2b.$
A. $S=4.$
B. $S=2.$
C. $S=8.$
D. $S=6.$
A. $S=4.$
B. $S=2.$
C. $S=8.$
D. $S=6.$
Ta có: $\int\limits_{0}^{\dfrac{\pi }{3}}{{{\sin }^{2}}x\cos xdx}=\int\limits_{0}^{\dfrac{\pi }{3}}{{{\sin }^{2}}xd\left( \sin x \right)}=\left. \dfrac{{{\sin }^{3}}x}{3} \right|_{0}^{\dfrac{\pi }{3}}=\dfrac{2\sqrt{3}}{16}\Rightarrow \left\{ \begin{aligned}
& a=0 \\
& b=2 \\
\end{aligned} \right.\Rightarrow S=4$.
& a=0 \\
& b=2 \\
\end{aligned} \right.\Rightarrow S=4$.
Đáp án A.