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Câu hỏi: Cho hàm số $f\left( x \right)$ có đạo hàm liên tục trên $\mathbb{R}$ thỏa mãn $f\left( 3 \right)=21$, $\int\limits_{0}^{3}{f\left( x \right)\text{d}x}=9$. Tính $I=\int\limits_{0}^{1}{x.{f}'\left( 3x \right)\text{d}x}$.
A. $I=15$.
B. $I=6$.
C. $I=12$.
D. $I=9$.
A. $I=15$.
B. $I=6$.
C. $I=12$.
D. $I=9$.
Ta có $I=\int\limits_{0}^{1}{x.{f}'\left( 3x \right)\text{d}x}=\dfrac{1}{9}\int\limits_{0}^{1}{3x.{f}'\left( 3x \right)\text{d}\left( 3x \right)}=\dfrac{1}{9}\int\limits_{0}^{3}{x.{f}'\left( x \right)\text{d}x}$.
Đặt $\left\{ \begin{aligned}
& u=x \\
& \text{d}v={f}'\left( x \right)\text{d}x \\
\end{aligned} \right.\Rightarrow \left\{ \begin{aligned}
& \text{d}u=\text{d}x \\
& v=f\left( x \right) \\
\end{aligned} \right.$.
Suy ra $\int\limits_{0}^{3}{x.{f}'\left( x \right)\text{d}x}=\left. x.f\left( x \right) \right|_{0}^{3}-\int\limits_{0}^{3}{f\left( x \right)\text{d}x}=3f\left( 3 \right)-9=3.21-9=54$.
Vậy $I=6$.
Đặt $\left\{ \begin{aligned}
& u=x \\
& \text{d}v={f}'\left( x \right)\text{d}x \\
\end{aligned} \right.\Rightarrow \left\{ \begin{aligned}
& \text{d}u=\text{d}x \\
& v=f\left( x \right) \\
\end{aligned} \right.$.
Suy ra $\int\limits_{0}^{3}{x.{f}'\left( x \right)\text{d}x}=\left. x.f\left( x \right) \right|_{0}^{3}-\int\limits_{0}^{3}{f\left( x \right)\text{d}x}=3f\left( 3 \right)-9=3.21-9=54$.
Vậy $I=6$.
Đáp án B.