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Câu hỏi: Nếu $\int\limits_{0}^{1}{\left[ {{f}^{2}}\left( x \right)-f\left( x \right) \right]}\text{d}x=5$ và $\int\limits_{0}^{1}{{{\left[ f\left( x \right)+1 \right]}^{2}}}\text{d}x=36$ thì $\int\limits_{0}^{1}{f\left( x \right)} \text{d}x$ bằng
A. 10.
B. 31.
C. 5.
D. 30.
A. 10.
B. 31.
C. 5.
D. 30.
Ta có $\int\limits_{0}^{1}{\left[ {{f}^{2}}\left( x \right)-f\left( x \right) \right]}\text{d}x=5$ $\Leftrightarrow \int\limits_{0}^{1}{{{f}^{2}}\left( x \right)\text{d}x-\int\limits_{0}^{1}{f\left( x \right)}}\text{d}x=5$ $\Leftrightarrow \int\limits_{0}^{1}{{{f}^{2}}\left( x \right)\text{d}x=\int\limits_{0}^{1}{f\left( x \right)}}\text{d}x+5 \left( 1 \right)$.
Lại có $\int\limits_{0}^{1}{{{\left[ f\left( x \right)+1 \right]}^{2}}}\text{d}x=36$ $\Leftrightarrow \int\limits_{0}^{1}{\left[ {{f}^{2}}\left( x \right)+2f\left( x \right)+1 \right]}\text{d}x=36$.
$\Leftrightarrow \int\limits_{0}^{1}{{{f}^{2}}\left( x \right)\text{d}x}+2\int\limits_{0}^{1}{f\left( x \right)\text{d}x+\int\limits_{0}^{1}{\text{d}x}}=36 \left( 2 \right)$. Thay $\left( 1 \right)$ vào $\left( 2 \right)$ ta được: $\int\limits_{0}^{1}{f\left( x \right)\text{d}x+5}+2\int\limits_{0}^{1}{f\left( x \right)\text{d}x}+1=36 \Leftrightarrow 3\int\limits_{0}^{1}{f\left( x \right)\text{d}x}=30$ $\Leftrightarrow \int\limits_{0}^{1}{f\left( x \right)\text{d}x=10}$. Vậy $\int\limits_{0}^{1}{f\left( x \right)} \text{d}x=10$.
Lại có $\int\limits_{0}^{1}{{{\left[ f\left( x \right)+1 \right]}^{2}}}\text{d}x=36$ $\Leftrightarrow \int\limits_{0}^{1}{\left[ {{f}^{2}}\left( x \right)+2f\left( x \right)+1 \right]}\text{d}x=36$.
$\Leftrightarrow \int\limits_{0}^{1}{{{f}^{2}}\left( x \right)\text{d}x}+2\int\limits_{0}^{1}{f\left( x \right)\text{d}x+\int\limits_{0}^{1}{\text{d}x}}=36 \left( 2 \right)$. Thay $\left( 1 \right)$ vào $\left( 2 \right)$ ta được: $\int\limits_{0}^{1}{f\left( x \right)\text{d}x+5}+2\int\limits_{0}^{1}{f\left( x \right)\text{d}x}+1=36 \Leftrightarrow 3\int\limits_{0}^{1}{f\left( x \right)\text{d}x}=30$ $\Leftrightarrow \int\limits_{0}^{1}{f\left( x \right)\text{d}x=10}$. Vậy $\int\limits_{0}^{1}{f\left( x \right)} \text{d}x=10$.
Đáp án A.