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Câu hỏi: Cho hàm số $f\left( x \right)$ có đạo hàm liên tục trên R. Biết $f\left( 3 \right)=1$ và $\int\limits_{0}^{1}{xf\left( 3x \right)dx}=1$, khi đó $\int\limits_{0}^{3}{{{x}^{2}}f'\left( x \right)dx}$ bằng
A. 3.
B. 7.
C. -9.
D. $\dfrac{25}{3}.$
A. 3.
B. 7.
C. -9.
D. $\dfrac{25}{3}.$
Đặt $t=3x\Rightarrow dt=3dx\Rightarrow dx=\dfrac{1}{3}dt.$
Suy ra $1=\int\limits_{0}^{1}{xf\left( 3x \right)dx}=\dfrac{1}{9}\int\limits_{0}^{3}{tf\left( t \right)dt}\Leftrightarrow \int\limits_{0}^{3}{tf\left( t \right)dt}=9$. Đặt $\left\{ \begin{aligned}
& u=f\left( t \right) \\
& dv=tdt \\
\end{aligned} \right.\Rightarrow \left\{ \begin{aligned}
& du=f'\left( t \right)dt \\
& v=\dfrac{{{t}^{2}}}{2} \\
\end{aligned} \right.$
$\Rightarrow \int\limits_{0}^{3}{tf\left( t \right)dt}=\dfrac{{{t}^{2}}}{2}\left. f\left( t \right) \right|_{0}^{3}-\int\limits_{0}^{3}{\dfrac{{{t}^{2}}}{2}f'\left( t \right)dt}=\dfrac{9}{2}f\left( 3 \right)-\dfrac{1}{2}\int\limits_{0}^{3}{{{t}^{2}}f'\left( t \right)dt}$
$\Leftrightarrow 9=\dfrac{9}{2}-\dfrac{1}{2}\int\limits_{0}^{3}{{{t}^{2}}f'\left( t \right)dt}\Leftrightarrow \int\limits_{0}^{3}{{{t}^{2}}f'\left( t \right)dt}=-9$. Vậy $\int\limits_{0}^{3}{{{x}^{2}}f'\left( x \right)dx}=-9$.
Suy ra $1=\int\limits_{0}^{1}{xf\left( 3x \right)dx}=\dfrac{1}{9}\int\limits_{0}^{3}{tf\left( t \right)dt}\Leftrightarrow \int\limits_{0}^{3}{tf\left( t \right)dt}=9$. Đặt $\left\{ \begin{aligned}
& u=f\left( t \right) \\
& dv=tdt \\
\end{aligned} \right.\Rightarrow \left\{ \begin{aligned}
& du=f'\left( t \right)dt \\
& v=\dfrac{{{t}^{2}}}{2} \\
\end{aligned} \right.$
$\Rightarrow \int\limits_{0}^{3}{tf\left( t \right)dt}=\dfrac{{{t}^{2}}}{2}\left. f\left( t \right) \right|_{0}^{3}-\int\limits_{0}^{3}{\dfrac{{{t}^{2}}}{2}f'\left( t \right)dt}=\dfrac{9}{2}f\left( 3 \right)-\dfrac{1}{2}\int\limits_{0}^{3}{{{t}^{2}}f'\left( t \right)dt}$
$\Leftrightarrow 9=\dfrac{9}{2}-\dfrac{1}{2}\int\limits_{0}^{3}{{{t}^{2}}f'\left( t \right)dt}\Leftrightarrow \int\limits_{0}^{3}{{{t}^{2}}f'\left( t \right)dt}=-9$. Vậy $\int\limits_{0}^{3}{{{x}^{2}}f'\left( x \right)dx}=-9$.
Đáp án C.