Google
Câu hỏi: Cho hàm số $y=f\left( x \right)$ liên tục và thỏa mãn $f\left( x \right)+2f\left( \dfrac{1}{x} \right)=3x$ với $x\in \left[ \dfrac{1}{2};2 \right]$. Tính $\int\limits_{\dfrac{1}{2}}^{2}{\dfrac{f\left( x \right)}{x}dx}.$
A. $-\dfrac{3}{2}.$
B. $\dfrac{9}{2}.$
C. $-\dfrac{9}{2}.$
D. $\dfrac{3}{2}.$
A. $-\dfrac{3}{2}.$
B. $\dfrac{9}{2}.$
C. $-\dfrac{9}{2}.$
D. $\dfrac{3}{2}.$
Xét $x\in \left[ \dfrac{1}{2};2 \right]$. Ta có: $f\left( x \right)+2f\left( \dfrac{1}{x} \right)=3x\Leftrightarrow \dfrac{f\left( x \right)}{x}+2\dfrac{f\left( \dfrac{1}{x} \right)}{x}=3$
Lấy tích phân $2$ vế ta được: $\int\limits_{\dfrac{1}{2}}^{2}{\dfrac{f\left( x \right)}{x}dx}+2\int\limits_{\dfrac{1}{2}}^{2}{\dfrac{f\left( \dfrac{1}{x} \right)}{x}dx}=\int\limits_{\dfrac{1}{2}}^{2}{3xdx}=\dfrac{9}{2}.$
Đặt $\dfrac{1}{x}=t\Rightarrow -\dfrac{1}{{{x}^{2}}}dx=dt\Leftrightarrow dx=-\dfrac{1}{{{t}^{2}}}dt$. Đổi cận: $\left\{ \begin{aligned}
& x=\dfrac{1}{2}\Rightarrow t=2 \\
& x=2\Rightarrow t=\dfrac{1}{2} \\
\end{aligned} \right.$
Khi đó: $2\int\limits_{\dfrac{1}{2}}^{2}{\dfrac{f\left( \dfrac{1}{x} \right)}{x}dx=}2\int\limits_{\dfrac{1}{2}}^{2}{\dfrac{f\left( t \right)}{t}dt=2}\int\limits_{\dfrac{1}{2}}^{2}{\dfrac{f\left( x \right)}{x}dx}\Rightarrow 3\int\limits_{\dfrac{1}{2}}^{2}{\dfrac{f\left( x \right)}{x}dx}=\dfrac{9}{2}\Leftrightarrow \int\limits_{\dfrac{1}{2}}^{2}{\dfrac{f\left( x \right)}{x}dx}=\dfrac{3}{2}$.
Lấy tích phân $2$ vế ta được: $\int\limits_{\dfrac{1}{2}}^{2}{\dfrac{f\left( x \right)}{x}dx}+2\int\limits_{\dfrac{1}{2}}^{2}{\dfrac{f\left( \dfrac{1}{x} \right)}{x}dx}=\int\limits_{\dfrac{1}{2}}^{2}{3xdx}=\dfrac{9}{2}.$
Đặt $\dfrac{1}{x}=t\Rightarrow -\dfrac{1}{{{x}^{2}}}dx=dt\Leftrightarrow dx=-\dfrac{1}{{{t}^{2}}}dt$. Đổi cận: $\left\{ \begin{aligned}
& x=\dfrac{1}{2}\Rightarrow t=2 \\
& x=2\Rightarrow t=\dfrac{1}{2} \\
\end{aligned} \right.$
Khi đó: $2\int\limits_{\dfrac{1}{2}}^{2}{\dfrac{f\left( \dfrac{1}{x} \right)}{x}dx=}2\int\limits_{\dfrac{1}{2}}^{2}{\dfrac{f\left( t \right)}{t}dt=2}\int\limits_{\dfrac{1}{2}}^{2}{\dfrac{f\left( x \right)}{x}dx}\Rightarrow 3\int\limits_{\dfrac{1}{2}}^{2}{\dfrac{f\left( x \right)}{x}dx}=\dfrac{9}{2}\Leftrightarrow \int\limits_{\dfrac{1}{2}}^{2}{\dfrac{f\left( x \right)}{x}dx}=\dfrac{3}{2}$.
Đáp án D.