Cho hàm số $y=f\left( x \right)$ có đạo hàm ${f}'\left( x \right)$...

Xét .
Đặt $\left\{ \begin{aligned}
& x=u \\
& {f}'\left( 2x \right)\text{d}x=\text{d}v \\
\end{aligned} \right.$$\Rightarrow \left\{ \begin{aligned}
& \text{d}x=\text{d}u \\
& \dfrac{1}{2}f\left( 2x \right)=v \\
\end{aligned} \right.I=\int\limits_{0}^{2}{x{f}'\left( 2x \right)\text{d}x}=\left. \dfrac{1}{2}xf\left( 2x \right) \right|_{0}^{2}-\dfrac{1}{2}\int\limits_{0}^{2}{f\left( 2x \right)\text{d}x}=\dfrac{1}{2}.2.f\left( 4 \right)-\dfrac{1}{4}\int\limits_{0}^{4}{f\left( x \right)\text{d}x}=8-\dfrac{1}{4}.6=\dfrac{13}{2}$.