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Câu hỏi: Cho hàm số $y=f\left( x \right)$ liên tục và thoả mãn $f\left( x \right)+2f\left( \dfrac{1}{x} \right)=3x$ với $x\in \left[ \dfrac{1}{2};2 \right]$.
Tính $\int\limits_{\dfrac{1}{2}}^{2}{\dfrac{f\left( x \right)}{x}\text{d}x}$.
A. $\dfrac{3}{2}$.
B. $-\dfrac{3}{2}$.
C. $\dfrac{9}{2}$.
D. $-\dfrac{9}{2}$.
Tính $\int\limits_{\dfrac{1}{2}}^{2}{\dfrac{f\left( x \right)}{x}\text{d}x}$.
A. $\dfrac{3}{2}$.
B. $-\dfrac{3}{2}$.
C. $\dfrac{9}{2}$.
D. $-\dfrac{9}{2}$.
Đặt $I=\int\limits_{\dfrac{1}{2}}^{2}{\dfrac{f\left( x \right)}{x}\text{d}x}$
Với $x\in \left[ \dfrac{1}{2};2 \right]$, $f\left( x \right)+2f\left( \dfrac{1}{x} \right)=3x$ $\Leftrightarrow \dfrac{f\left( x \right)}{x}+2\dfrac{f\left( \dfrac{1}{x} \right)}{x}=3$.
$\Rightarrow \int\limits_{\dfrac{1}{2}}^{2}{\dfrac{f\left( x \right)}{x}}\text{d}x+2\int\limits_{\dfrac{1}{2}}^{2}{\dfrac{f\left( \dfrac{1}{x} \right)}{x}\text{d}x}=\int\limits_{\dfrac{1}{2}}^{2}{3}\text{d}x (1)$
Đặt $t=\dfrac{1}{x}\Rightarrow \text{d}t=-\dfrac{1}{{{x}^{2}}}\text{d}x$ $\Rightarrow -\dfrac{1}{t}\text{d}t=\dfrac{1}{x}\text{d}x$.
$2\int\limits_{\dfrac{1}{2}}^{2}{\dfrac{f\left( \dfrac{1}{x} \right)}{x}\text{d}x}=2\int\limits_{\dfrac{1}{2}}^{2}{\dfrac{f\left( t \right)}{t}}\text{d}t=2I$.
$\left( 1 \right)\Rightarrow 3I=\int\limits_{\dfrac{1}{2}}^{2}{3}\text{d}x\Rightarrow I=\dfrac{3}{2}.$
Với $x\in \left[ \dfrac{1}{2};2 \right]$, $f\left( x \right)+2f\left( \dfrac{1}{x} \right)=3x$ $\Leftrightarrow \dfrac{f\left( x \right)}{x}+2\dfrac{f\left( \dfrac{1}{x} \right)}{x}=3$.
$\Rightarrow \int\limits_{\dfrac{1}{2}}^{2}{\dfrac{f\left( x \right)}{x}}\text{d}x+2\int\limits_{\dfrac{1}{2}}^{2}{\dfrac{f\left( \dfrac{1}{x} \right)}{x}\text{d}x}=\int\limits_{\dfrac{1}{2}}^{2}{3}\text{d}x (1)$
Đặt $t=\dfrac{1}{x}\Rightarrow \text{d}t=-\dfrac{1}{{{x}^{2}}}\text{d}x$ $\Rightarrow -\dfrac{1}{t}\text{d}t=\dfrac{1}{x}\text{d}x$.
$2\int\limits_{\dfrac{1}{2}}^{2}{\dfrac{f\left( \dfrac{1}{x} \right)}{x}\text{d}x}=2\int\limits_{\dfrac{1}{2}}^{2}{\dfrac{f\left( t \right)}{t}}\text{d}t=2I$.
$\left( 1 \right)\Rightarrow 3I=\int\limits_{\dfrac{1}{2}}^{2}{3}\text{d}x\Rightarrow I=\dfrac{3}{2}.$
Đáp án A.