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Câu hỏi: Cho hàm số $f\left( x \right)$ liên tục trên $\mathbb{R}$. Biết $\int\limits_{1}^{{{e}^{3}}}{\dfrac{f\left( \ln x \right)}{x}\text{d}x}=7$, $\int\limits_{0}^{\dfrac{\pi }{2}}{f\left( \cos x \right)\sin x\text{d}x}=3$. Giá trị của $\int\limits_{1}^{3}{\left[ f\left( x \right)+2x \right]\text{d}x}$ bằng
A. $10$.
B. $-10$.
C. $15$.
D. $12$.
A. $10$.
B. $-10$.
C. $15$.
D. $12$.
Ta có $\int\limits_{1}^{{{e}^{3}}}{\dfrac{f\left( \ln x \right)}{x}\text{d}x}=\int\limits_{1}^{{{e}^{3}}}{f\left( \ln x \right)\text{d}\left( \ln x \right)}=\int\limits_{0}^{3}{f\left( t \right)\text{d}t}=\int\limits_{0}^{3}{f\left( x \right)\text{d}x}=7$.
Ta có $\int\limits_{0}^{\dfrac{\pi }{2}}{f\left( \cos x \right)\sin x\text{d}x}=-\int\limits_{0}^{\dfrac{\pi }{2}}{f\left( \cos x \right)\text{d}\left( \cos x \right)}=-\int\limits_{1}^{0}{f\left( u \right)\text{d}u}=\int\limits_{0}^{1}{f\left( x \right)\text{d}x}=3$.
Khi đó $\int\limits_{1}^{3}{\left[ f\left( x \right)+2x \right]\text{d}x}=\int\limits_{0}^{3}{f\left( x \right)\text{d}x}-\int\limits_{0}^{1}{f\left( x \right)\text{d}x}+\int\limits_{1}^{3}{2x\text{d}x}=7-3+8=12$.
Ta có $\int\limits_{0}^{\dfrac{\pi }{2}}{f\left( \cos x \right)\sin x\text{d}x}=-\int\limits_{0}^{\dfrac{\pi }{2}}{f\left( \cos x \right)\text{d}\left( \cos x \right)}=-\int\limits_{1}^{0}{f\left( u \right)\text{d}u}=\int\limits_{0}^{1}{f\left( x \right)\text{d}x}=3$.
Khi đó $\int\limits_{1}^{3}{\left[ f\left( x \right)+2x \right]\text{d}x}=\int\limits_{0}^{3}{f\left( x \right)\text{d}x}-\int\limits_{0}^{1}{f\left( x \right)\text{d}x}+\int\limits_{1}^{3}{2x\text{d}x}=7-3+8=12$.
Đáp án D.