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Câu hỏi: Biết $\int\limits_{0}^{\dfrac{\pi }{2}}{\left[ f\left( x \right)+2\cos x \right] \text{d}x}=3$. Khi đó $\int\limits_{0}^{\dfrac{\pi }{2}}{f\left( x \right)\text{d}x}$ bằng
A. $3$.
B. $2$.
C. $1$.
D. $4$.
A. $3$.
B. $2$.
C. $1$.
D. $4$.
Ta có $\int\limits_{0}^{\dfrac{\pi }{2}}{\left[ f\left( x \right)+2\cos x \right] \text{d}x}=3$ $\Leftrightarrow \int\limits_{0}^{\dfrac{\pi }{2}}{f\left( x \right) \text{d}x}+2\int\limits_{0}^{\dfrac{\pi }{2}}{\cos x \text{d}x}=3$
$\Leftrightarrow \int\limits_{0}^{\dfrac{\pi }{2}}{f\left( x \right) \text{d}x}+\left. 2\sin x \right|_{0}^{\dfrac{\pi }{2}}=3\Leftrightarrow \int\limits_{0}^{\dfrac{\pi }{2}}{f\left( x \right) \text{d}x}+2=3\Leftrightarrow \int\limits_{0}^{\dfrac{\pi }{2}}{f\left( x \right) \text{d}x}=1.$
$\Leftrightarrow \int\limits_{0}^{\dfrac{\pi }{2}}{f\left( x \right) \text{d}x}+\left. 2\sin x \right|_{0}^{\dfrac{\pi }{2}}=3\Leftrightarrow \int\limits_{0}^{\dfrac{\pi }{2}}{f\left( x \right) \text{d}x}+2=3\Leftrightarrow \int\limits_{0}^{\dfrac{\pi }{2}}{f\left( x \right) \text{d}x}=1.$
Đáp án C.
