Cho hàm số

C

- Hướng dẫn giải

Đáp án C

Ta có $I=\underset{\frac{1}{2}}{\overset{2}{\int }}\frac{f\left(x\right)}{x}dx=I=\underset{\frac{1}{2}}{\overset{2}{\int }}\frac{1}{x}\left[3x-2f\left(\frac{1}{x}\right)\right]dx=3\underset{\frac{1}{2}}{\overset{2}{\int }}dx-2\underset{\frac{1}{2}}{\overset{2}{\int }}\frac{1}{x}f\left(\frac{1}{x}\right)dx$ 

Đặt $t=\frac{1}{x}\iff dt=-\frac{dx}{x^2}\iff dx=-\frac{dt}{t^2}$ và $\left\{\begin{array}{l}x=\frac{1}{2}\Rightarrow t=2\\ x=2\Rightarrow t=\frac{1}{2}\end{array}\right..$ 

Suy ra $\underset{\frac{1}{2}}{\overset{2}{\int }}\frac{1}{x}f\left(\frac{1}{x}\right)dx=\underset{\frac{1}{2}}{\overset{2}{\int }}\frac{f\left(t\right)}{t}dt$

Vậy $I=3\underset{\frac{1}{2}}{\overset{2}{\int }}dx-2\underset{\frac{1}{2}}{\overset{2}{\int }}\frac{f\left(t\right)}{t}dt=\frac{9}{2}-2\underset{\frac{1}{2}}{\overset{2}{\int }}\frac{f\left(t\right)}{t}dt\Rightarrow 3I=\frac{9}{2}\iff I=\frac{3}{2}$