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Câu hỏi: Biết rằng $f,g$ là hai hàm liên tục trên $\left[ 1;3 \right],\int\limits_{1}^{3}{\left[ f\left( x \right)+3g\left( x \right) \right]\text{d}x=10}$ và $\int\limits_{1}^{3}{\left[ 2f\left( x \right)-g\left( x \right) \right]\text{d}x=6}$. Tính $\int\limits_{1}^{3}{\left[ f\left( x \right)+g\left( x \right) \right]\text{d}x}.$
A. $7$.
B. $8$.
C. $6$.
D. $9$.
A. $7$.
B. $8$.
C. $6$.
D. $9$.
Ta có: $\left\{ \begin{aligned}
& \int\limits_{1}^{3}{f\left( x \right)\text{d}x+3\int\limits_{1}^{3}{g\left( x \right)\text{d}x=10}} \\
& 2\int\limits_{1}^{3}{f\left( x \right)\text{d}x-\int\limits_{1}^{3}{g\left( x \right)\text{d}x=6}} \\
\end{aligned} \right.\Leftrightarrow \left\{ \begin{aligned}
& \int\limits_{1}^{3}{f\left( x \right)\text{d}x}=4 \\
& \int\limits_{1}^{3}{g\left( x \right)\text{d}x}=2 \\
\end{aligned} \right.\Rightarrow \int\limits_{1}^{3}{\left[ f\left( x \right)+g\left( x \right) \right]\text{d}x}=6$.
& \int\limits_{1}^{3}{f\left( x \right)\text{d}x+3\int\limits_{1}^{3}{g\left( x \right)\text{d}x=10}} \\
& 2\int\limits_{1}^{3}{f\left( x \right)\text{d}x-\int\limits_{1}^{3}{g\left( x \right)\text{d}x=6}} \\
\end{aligned} \right.\Leftrightarrow \left\{ \begin{aligned}
& \int\limits_{1}^{3}{f\left( x \right)\text{d}x}=4 \\
& \int\limits_{1}^{3}{g\left( x \right)\text{d}x}=2 \\
\end{aligned} \right.\Rightarrow \int\limits_{1}^{3}{\left[ f\left( x \right)+g\left( x \right) \right]\text{d}x}=6$.
Đáp án C.