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Câu hỏi: Cho hàm số $f\left( x \right)=\left\{ \begin{aligned}
& {{x}^{2}}-5x+3\text{ khi }x\ge 7 \\
& 2x+3\text{ khi }x<7 \\
\end{aligned} \right. $. Tích phân $ \int\limits_{0}^{\ln 4}{f\left( 2{{e}^{x}}+3 \right){{e}^{x}}dx}$ bằng
A. $\dfrac{1148}{3}$.
B. $\dfrac{220}{3}$.
C. $\dfrac{115}{3}$.
D. $\dfrac{287}{3}$.
& {{x}^{2}}-5x+3\text{ khi }x\ge 7 \\
& 2x+3\text{ khi }x<7 \\
\end{aligned} \right. $. Tích phân $ \int\limits_{0}^{\ln 4}{f\left( 2{{e}^{x}}+3 \right){{e}^{x}}dx}$ bằng
A. $\dfrac{1148}{3}$.
B. $\dfrac{220}{3}$.
C. $\dfrac{115}{3}$.
D. $\dfrac{287}{3}$.
Xét tích phân $I=\int\limits_{0}^{\ln 4}{f\left( 2{{e}^{x}}+3 \right){{e}^{x}}dx}$. Đặt $t=2{{e}^{x}}+3\Rightarrow dt=2{{e}^{x}}dx$ hay ${{e}^{x}}dx=\dfrac{1}{2}dt$.
Đổi cận $x=0\Rightarrow t=5$ ; $x=\ln 4\Rightarrow t=11$. Khi đó
$I=\dfrac{1}{2}\int\limits_{5}^{11}{f\left( t \right)dt}=\dfrac{1}{2}\int\limits_{5}^{11}{f\left( x \right)dx}=\dfrac{1}{2}\left( \int\limits_{5}^{7}{f\left( x \right)dx}+\int\limits_{7}^{11}{f\left( x \right)dx} \right)=\dfrac{1}{2}\left[ \int\limits_{5}^{7}{\left( 2x+3 \right)dx}+\int\limits_{7}^{11}{\left( {{x}^{2}}-5x+3 \right)dx} \right]$
$=\dfrac{1}{2}\left[ \left. \left( {{x}^{2}}+3x \right) \right|_{5}^{7}+\left. \left( \dfrac{{{x}^{3}}}{3}-\dfrac{5{{x}^{2}}}{2}+3x \right) \right|_{7}^{11} \right]=\dfrac{1}{2}\left( 30+\dfrac{484}{3} \right)=\dfrac{287}{3}$.
Vậy $\int\limits_{0}^{\ln 4}{f\left( 2{{e}^{x}}+3 \right){{e}^{x}}dx}=\dfrac{287}{3}$.
Đổi cận $x=0\Rightarrow t=5$ ; $x=\ln 4\Rightarrow t=11$. Khi đó
$I=\dfrac{1}{2}\int\limits_{5}^{11}{f\left( t \right)dt}=\dfrac{1}{2}\int\limits_{5}^{11}{f\left( x \right)dx}=\dfrac{1}{2}\left( \int\limits_{5}^{7}{f\left( x \right)dx}+\int\limits_{7}^{11}{f\left( x \right)dx} \right)=\dfrac{1}{2}\left[ \int\limits_{5}^{7}{\left( 2x+3 \right)dx}+\int\limits_{7}^{11}{\left( {{x}^{2}}-5x+3 \right)dx} \right]$
$=\dfrac{1}{2}\left[ \left. \left( {{x}^{2}}+3x \right) \right|_{5}^{7}+\left. \left( \dfrac{{{x}^{3}}}{3}-\dfrac{5{{x}^{2}}}{2}+3x \right) \right|_{7}^{11} \right]=\dfrac{1}{2}\left( 30+\dfrac{484}{3} \right)=\dfrac{287}{3}$.
Vậy $\int\limits_{0}^{\ln 4}{f\left( 2{{e}^{x}}+3 \right){{e}^{x}}dx}=\dfrac{287}{3}$.
Đáp án D.