Google
Câu hỏi: Cho hàm số $y=f\left( x \right)=\left\{ \begin{aligned}
& {{x}^{2}}+3{{x}^{2}};x\ge 1 \\
& 5-x ;x<1 \\
\end{aligned} \right. $. Tính $ I=2\int\limits_{0}^{\dfrac{\pi }{2}}{f\left( \sin x \right)\cos x\text{d}x+3\int\limits_{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{{x}^{2}};x\ge 1 \\
& 5-x ;x<1 \\
\end{aligned} \right. $. Tính $ I=2\int\limits_{0}^{\dfrac{\pi }{2}}{f\left( \sin x \right)\cos x\text{d}x+3\int\limits_{0}^{1}{f\left( 3-2x \right)\text{d}x}}$.
A. $I=\dfrac{71}{6}$.
B. $I=31$.
C. $I=32$.
D. $I=\dfrac{32}{3}$.
Xét tích phân ${{I}_{1}}=\int\limits_{0}^{\dfrac{\pi }{2}}{f\left( \sin x \right)\cos x\text{d}x}$.Đặt $t=sinx\Rightarrow \text{d}t=\cos x\text{d}x$
Đổi cận
Ta có ${{I}_{1}}=\int\limits_{0}^{1}{f\left( t \right)\text{d}t=}\int\limits_{0}^{1}{f\left( x \right)\text{d}x}=\int\limits_{0}^{1}{\left( 5-x \right)\text{d}x=}\left. \left( 5x-\dfrac{{{x}^{2}}}{2} \right) \right|_{0}^{1}=\dfrac{9}{2}$
Xét tích phân ${{I}_{2}}=\int\limits_{0}^{1}{f\left( 3-2x \right)\text{d}x}$.Đặt $t=3-2x\Rightarrow \text{d}t=-2\text{d}x\Rightarrow \text{d}x=\dfrac{-\text{d}t}{2}$
Đổi cận
Ta có ${{I}_{2}}=\int\limits_{0}^{1}{f\left( 3-2x \right)\text{d}x}=\dfrac{1}{2}\int\limits_{1}^{3}{f\left( t \right)\text{d}t=}\dfrac{1}{2}\int\limits_{1}^{3}{f\left( x \right)\text{d}x=}\dfrac{1}{2}\int\limits_{1}^{3}{\left( {{x}^{2}}+3 \right)\text{d}x=}\dfrac{1}{2}\left. \left( \dfrac{{{x}^{3}}}{3}+3x \right) \right|_{1}^{3}=\dfrac{1}{2}\left( 18-\dfrac{10}{3} \right)=\dfrac{22}{3}$
Vậy $I=2\int\limits_{0}^{\dfrac{\pi }{2}}{f\left( \sin x \right)\cos x\text{d}x+3\int\limits_{0}^{1}{f\left( 3-2x \right)\text{d}x}}=9+22=31$.
Đổi cận
| $x$ | $0$ | $\dfrac{\pi }{2}$ |
| $t$ | $0$ | $1$ |
Xét tích phân ${{I}_{2}}=\int\limits_{0}^{1}{f\left( 3-2x \right)\text{d}x}$.Đặt $t=3-2x\Rightarrow \text{d}t=-2\text{d}x\Rightarrow \text{d}x=\dfrac{-\text{d}t}{2}$
Đổi cận
| $x$ | $0$ | $1$ |
| $t$ | 3 | $1$ |
Vậy $I=2\int\limits_{0}^{\dfrac{\pi }{2}}{f\left( \sin x \right)\cos x\text{d}x+3\int\limits_{0}^{1}{f\left( 3-2x \right)\text{d}x}}=9+22=31$.
Đáp án B.