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Câu hỏi: Biết $F\left( x \right)$ là một nguyên hàm của hàm số $f\left( x \right)$ trên đoạn $\left[ -1;0 \right],F\left( -1 \right)=-1,F\left( 0 \right)=0$ và $\int\limits_{-1}^{0}{{{2}^{3\text{x}}}F\left( x \right)d\text{x}}=-1.$ Tính $I=\int\limits_{-1}^{0}{{{2}^{3\text{x}}}f\left( x \right)d\text{x}}$
A. $I=\dfrac{1}{8}-3\ln 2$
B. $I=\dfrac{1}{8}+\ln 2$
C. $I=\dfrac{1}{8}+3\ln 2$
D. $I=-\dfrac{1}{8}+3\ln 2$
A. $I=\dfrac{1}{8}-3\ln 2$
B. $I=\dfrac{1}{8}+\ln 2$
C. $I=\dfrac{1}{8}+3\ln 2$
D. $I=-\dfrac{1}{8}+3\ln 2$
Đặt $\left\{ \begin{aligned}
& u=F\left( x \right) \\
& dv={{2}^{3x}}dx \\
\end{aligned} \right.\Rightarrow \left\{ \begin{aligned}
& du=f\left( x \right)d\text{x} \\
& v=\dfrac{{{2}^{3x}}}{3\ln 2} \\
\end{aligned} \right.\Rightarrow \int\limits_{-1}^{0}{{{2}^{3x}}.F\left( x \right)d\text{x}}=\left. \dfrac{F\left( x \right){{2}^{3x}}}{3\ln 2} \right|_{-1}^{0}-\int\limits_{-1}^{0}{\dfrac{{{2}^{3x}}.f\left( x \right)d\text{x}}{3\ln 2}}$
$=-3\ln 2=-\dfrac{F\left( -1 \right)}{8}-I\Rightarrow I=\dfrac{1}{8}+3\ln 2$
& u=F\left( x \right) \\
& dv={{2}^{3x}}dx \\
\end{aligned} \right.\Rightarrow \left\{ \begin{aligned}
& du=f\left( x \right)d\text{x} \\
& v=\dfrac{{{2}^{3x}}}{3\ln 2} \\
\end{aligned} \right.\Rightarrow \int\limits_{-1}^{0}{{{2}^{3x}}.F\left( x \right)d\text{x}}=\left. \dfrac{F\left( x \right){{2}^{3x}}}{3\ln 2} \right|_{-1}^{0}-\int\limits_{-1}^{0}{\dfrac{{{2}^{3x}}.f\left( x \right)d\text{x}}{3\ln 2}}$
$=-3\ln 2=-\dfrac{F\left( -1 \right)}{8}-I\Rightarrow I=\dfrac{1}{8}+3\ln 2$
Đáp án C.