Cho hàm số \(y=f\left( x \right)\) liên tục và thoả mãn \(f\left( x \right)+2f\left( \frac{1}{x} \right)=3x\) với \(x\in \left[ \frac{1}{2};2 \right]\). Tính \(\int\limits_{\frac{1...

* Hướng dẫn giải

Đặt \(I=\int\limits_{\frac{1}{2}}^{2}{\frac{f\left( x \right)}{x}\text{d}x}\)

Với \(x\in \left[ \frac{1}{2};2 \right], f\left( x \right)+2f\left( \frac{1}{x} \right)=3x\Leftrightarrow \frac{f\left( x \right)}{x}+2\frac{f\left( \frac{1}{x} \right)}{x}=3\) .

\(\Rightarrow \int\limits_{\frac{1}{2}}^{2}{\frac{f\left( x \right)}{x}}\text{d}x+2\int\limits_{\frac{1}{2}}^{2}{\frac{f\left( \frac{1}{x} \right)}{x}\text{d}x}=\int\limits_{\frac{1}{2}}^{2}{3}\text{d}x\,\,\,\,\,(1)\)

Đặt \(t=\frac{1}{x}\Rightarrow \text{d}t=-\frac{1}{{{x}^{2}}}\text{d}x\Rightarrow -\frac{1}{t}\text{d}t=\frac{1}{x}\text{d}x\).

\(2\int\limits_{\frac{1}{2}}^{2}{\frac{f\left( \frac{1}{x} \right)}{x}\text{d}x}=2\int\limits_{\frac{1}{2}}^{2}{\frac{f\left( t \right)}{t}}\text{d}t=2I\)

\(\left( 1 \right)\Rightarrow 3I=\int\limits_{\frac{1}{2}}^{2}{3}\text{d}x\Rightarrow I=\frac{3}{2}.\)