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Câu hỏi: Cho $\int\limits_{0}^{\ln 2}{\left( 2f\left( x \right)+{{e}^{x}} \right)dx}=5$. Khi đó $\int\limits_{0}^{\ln 2}{f\left( x \right)dx}$ bằng
A. $3$.
B. $1$.
C. $2$.
D. $\dfrac{5}{2}$.
A. $3$.
B. $1$.
C. $2$.
D. $\dfrac{5}{2}$.
Ta có
$\begin{aligned}
& \int\limits_{0}^{\ln 2}{\left( 2f\left( x \right)+{{e}^{x}} \right)dx}=2\int\limits_{0}^{\ln 2}{f\left( x \right)dx}+\int\limits_{0}^{\ln 2}{{{e}^{x}}dx}=2\int\limits_{0}^{\ln 2}{f\left( x \right)dx}+\left. {{e}^{x}} \right|_{0}^{\ln 2}=2\int\limits_{0}^{\ln 2}{f\left( x \right)dx}+1 \\
& \Rightarrow \int\limits_{0}^{\ln 2}{f\left( x \right)dx}=2 \\
\end{aligned}$.
$\begin{aligned}
& \int\limits_{0}^{\ln 2}{\left( 2f\left( x \right)+{{e}^{x}} \right)dx}=2\int\limits_{0}^{\ln 2}{f\left( x \right)dx}+\int\limits_{0}^{\ln 2}{{{e}^{x}}dx}=2\int\limits_{0}^{\ln 2}{f\left( x \right)dx}+\left. {{e}^{x}} \right|_{0}^{\ln 2}=2\int\limits_{0}^{\ln 2}{f\left( x \right)dx}+1 \\
& \Rightarrow \int\limits_{0}^{\ln 2}{f\left( x \right)dx}=2 \\
\end{aligned}$.
Đáp án C.