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Câu hỏi: Cho $\underset{0}{\overset{1}{\mathop \int }} \left[ f\left( x \right)+g\left( x \right) \right]dx=2$ và $\underset{0}{\overset{1}{\mathop \int }} \left[ 2f\left( x \right)+g\left( x \right) \right]dx=3,$ khi đó $\int\limits_{0}^{1}{f\left( x \right)dx}$ bằng
A. 5.
B. 2.
C. 1.
D. $-3.$
A. 5.
B. 2.
C. 1.
D. $-3.$
Ta có $\underset{0}{\overset{1}{\mathop \int }} \left[ f\left( x \right)+g\left( x \right) \right]dx=2\Leftrightarrow \underset{0}{\overset{1}{\mathop \int }} f\left( x \right)dx+\underset{0}{\overset{1}{\mathop \int }} g\left( x \right)dx=2\left( 1 \right)$
Và $\underset{0}{\overset{1}{\mathop \int }} \left[ 2f\left( x \right)+g\left( x \right) \right]dx=3\Leftrightarrow 2\underset{0}{\overset{1}{\mathop \int }} f\left( x \right)dx+\underset{0}{\overset{1}{\mathop \int }} g\left( x \right)dx=3\left( 2 \right)$
Lấy $\left( 2 \right)-\left( 1 \right),$ ta được $\int\limits_{0}^{1}{f\left( x \right)dx}=1.$
Và $\underset{0}{\overset{1}{\mathop \int }} \left[ 2f\left( x \right)+g\left( x \right) \right]dx=3\Leftrightarrow 2\underset{0}{\overset{1}{\mathop \int }} f\left( x \right)dx+\underset{0}{\overset{1}{\mathop \int }} g\left( x \right)dx=3\left( 2 \right)$
Lấy $\left( 2 \right)-\left( 1 \right),$ ta được $\int\limits_{0}^{1}{f\left( x \right)dx}=1.$
Đáp án C.