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Câu hỏi: Cho hàm số $f\left( x \right)$ liên tục trên $\mathbb{R}$. Nếu $\int\limits_{1}^{3}{f\left( 2x-1 \right)\text{d}x}=3$ thì $\int\limits_{1}^{5}{f\left( x \right)\text{d}x}$ bằng
A. $3$
B. $\dfrac{3}{2}$
C. $1$
D. $6$.
A. $3$
B. $\dfrac{3}{2}$
C. $1$
D. $6$.
Đặt $t=2x-1\Rightarrow dt=2dx\Rightarrow dx=\dfrac{1}{2}dt$.
Đổi cận $x=1\Rightarrow t=1; x=3\Rightarrow t=5$.
Ta có $\int\limits_{1}^{3}{f\left( 2x-1 \right)\text{d}x}=\dfrac{1}{2}\int\limits_{1}^{5}{f\left( t \right)\text{dt} \text{=}}\dfrac{1}{2}\int\limits_{1}^{5}{f\left( x \right)\text{dx} \text{=} }3\Rightarrow \int\limits_{1}^{5}{f\left( x \right)\text{dx} \text{=} }6$.
Đổi cận $x=1\Rightarrow t=1; x=3\Rightarrow t=5$.
Ta có $\int\limits_{1}^{3}{f\left( 2x-1 \right)\text{d}x}=\dfrac{1}{2}\int\limits_{1}^{5}{f\left( t \right)\text{dt} \text{=}}\dfrac{1}{2}\int\limits_{1}^{5}{f\left( x \right)\text{dx} \text{=} }3\Rightarrow \int\limits_{1}^{5}{f\left( x \right)\text{dx} \text{=} }6$.
Đáp án D.