Xét \(I=\int\limits^1_0x.f\left(3x\right)dx\)
Đặt \(3x=u\Rightarrow dx=\dfrac{1}{3}du\) ; \(\left\{{}\begin{matrix}x=0\Rightarrow u=0\\x=1\Rightarrow u=3\end{matrix}\right.\)
\(\Rightarrow I=\dfrac{1}{9}\int\limits^3_0u.f\left(u\right)du=\dfrac{1}{9}\int\limits^3_0x.f\left(x\right)dx=1\)
\(\Rightarrow J=\int\limits^3_0x.f\left(x\right)dx=9\)
Xét J, đặt \(\left\{{}\begin{matrix}u=f\left(x\right)\\dv=x.dx\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}du=f'\left(x\right)dx\\v=\dfrac{x^2}{2}\end{matrix}\right.\)
\(\Rightarrow J=\dfrac{x^2}{2}.f\left(x\right)|^3_0-\dfrac{1}{2}\int\limits^3_0x^2.f'\left(x\right)dx=\dfrac{9}{2}-\dfrac{1}{2}\int\limits^3_0x^2.f'\left(x\right)dx\)
\(\Rightarrow\int\limits^3_0x^2.f'\left(x\right)dx=9-2J=-9\)
