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Câu hỏi: Cho hàm số $y=f\left( x \right)=\left\{ \begin{aligned}
& {{x}^{2}}+3\text{ }khi x\ge 1 \\
& 5-x\text{ khi }x<1 \\
\end{aligned} \right. $. Tính $ I=2\int_{0}^{\dfrac{\pi }{2}}{f\left( \sin x \right)\cos x\text{d}x}+3\int_{0}^{1}{f\left( 3-2x \right)\text{d}x}$
A. $I=\dfrac{71}{6}$.
B. $I=31$.
C. $I=32$.
D. $I=\dfrac{32}{3}$.
& {{x}^{2}}+3\text{ }khi x\ge 1 \\
& 5-x\text{ khi }x<1 \\
\end{aligned} \right. $. Tính $ I=2\int_{0}^{\dfrac{\pi }{2}}{f\left( \sin x \right)\cos x\text{d}x}+3\int_{0}^{1}{f\left( 3-2x \right)\text{d}x}$
A. $I=\dfrac{71}{6}$.
B. $I=31$.
C. $I=32$.
D. $I=\dfrac{32}{3}$.
$I=2\int_{0}^{\dfrac{\pi }{2}}{f\left( \sin x \right)\cos x\text{d}x}+3\int_{0}^{1}{f\left( 3-2x \right)\text{d}x}\text{ =}2\int_{0}^{\dfrac{\pi }{2}}{f\left( \sin x \right)\text{d}\left( \sin x \right)}-\dfrac{3}{2}\int_{0}^{1}{f\left( 3-2x \right)\text{d}\left( 3-2x \right)}$
$\text{=}2\int_{0}^{1}{f\left( x \right)\text{d}x}+\dfrac{3}{2}\int_{1}^{3}{f\left( x \right)\text{d}x} =2\int_{0}^{1}{\left( 5-x \right)\text{d}x}+\dfrac{3}{2}\int_{1}^{3}{\left( {{x}^{2}}+3 \right)\text{d}x}$
$\text{ }=9+22=31$
$\text{=}2\int_{0}^{1}{f\left( x \right)\text{d}x}+\dfrac{3}{2}\int_{1}^{3}{f\left( x \right)\text{d}x} =2\int_{0}^{1}{\left( 5-x \right)\text{d}x}+\dfrac{3}{2}\int_{1}^{3}{\left( {{x}^{2}}+3 \right)\text{d}x}$
$\text{ }=9+22=31$
Đáp án B.