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Câu hỏi: Cho $f, g$ là hai hàm liên tục trên $\left[ 1 ; 3 \right]$ thỏa mãn điều kiện $\int\limits_{1}^{3}{\left[ f\left( x \right)+3g\left( x \right) \right] \text{d}x}=10$ đồng thời $\int\limits_{1}^{3}{\left[ 2f\left( x \right)-g\left( x \right) \right] \text{d}x}=6$. Tính $\int\limits_{1}^{3}{\left[ f\left( x \right)+g\left( x \right) \right] \text{d}x}$.
A. $9$.
B. $6$.
C. $7$.
D. $8$.
A. $9$.
B. $6$.
C. $7$.
D. $8$.
Đặt $a=\int\limits_{1}^{3}{f\left( x \right) \text{d}x}$, $b=\int\limits_{1}^{3}{g\left( x \right) \text{d}x}$. Khi đó $\int\limits_{1}^{3}{\left[ f\left( x \right)+3g\left( x \right) \right] \text{d}x}=10$ $\Leftrightarrow a+3b=10$, $\int\limits_{1}^{3}{\left[ 2f\left( x \right)-g\left( x \right) \right] \text{d}x}=6$ $\Leftrightarrow 2a-b=6$.
Do đó: $\left\{ \begin{aligned}
& a+3b=10 \\
& 2a-b=6 \\
\end{aligned} \right.$$\Leftrightarrow \left\{ \begin{aligned}
& a=4 \\
& b=2 \\
\end{aligned} \right. $. Vậy $ \int\limits_{1}^{3}{\left[ f\left( x \right)+g\left( x \right) \right] \text{d}x} $ $ =a+b=6$.
Do đó: $\left\{ \begin{aligned}
& a+3b=10 \\
& 2a-b=6 \\
\end{aligned} \right.$$\Leftrightarrow \left\{ \begin{aligned}
& a=4 \\
& b=2 \\
\end{aligned} \right. $. Vậy $ \int\limits_{1}^{3}{\left[ f\left( x \right)+g\left( x \right) \right] \text{d}x} $ $ =a+b=6$.
Đáp án B.