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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 $\int\limits_{0}^{\dfrac{\pi }{3}}{{{\sin }^{2}}x\cos xdx}=\int\limits_{0}^{\dfrac{\pi }{3}}{{{\sin }^{2}}xd\left( \sin x \right)}=\dfrac{{{\sin }^{2}}x}{3}\left| \begin{aligned}
& ^{\dfrac{\pi }{3}} \\
& _{0} \\
\end{aligned} \right.=\dfrac{2\sqrt{3}}{16}\Rightarrow \left\{ \begin{aligned}
& a=0 \\
& b=2 \\
\end{aligned} \right.\Rightarrow S=4$
& ^{\dfrac{\pi }{3}} \\
& _{0} \\
\end{aligned} \right.=\dfrac{2\sqrt{3}}{16}\Rightarrow \left\{ \begin{aligned}
& a=0 \\
& b=2 \\
\end{aligned} \right.\Rightarrow S=4$
Đáp án A.