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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 $I=\int_{0}^{1} x . f^{\prime}(3 x) \mathrm{d} x=\dfrac{1}{9} \int_{0}^{1} 3 x . f^{\prime}(3 x) \mathrm{d}(3 x)=\dfrac{1}{9} \int_{0}^{3} x \cdot f^{\prime}(x) \mathrm{d} x$.
Đặt $\left\{\begin{array}{l}u=x \\ \mathrm{~d} v=f^{\prime}(x) \mathrm{d} x\end{array} \Rightarrow\left\{\begin{array}{l}\mathrm{d} u=\mathrm{d} x \\ v=f(x)\end{array}\right.\right.$.
Suy ra $\int_{0}^{3} x \cdot f^{\prime}(x) \mathrm{d} x=\left.x \cdot f(x)\right|_{0} ^{3}-\int_{0}^{3} f(x) \mathrm{d} x=3 f(3)-9=3.21-9=54$.
Vậy $I=6$.
Đặt $\left\{\begin{array}{l}u=x \\ \mathrm{~d} v=f^{\prime}(x) \mathrm{d} x\end{array} \Rightarrow\left\{\begin{array}{l}\mathrm{d} u=\mathrm{d} x \\ v=f(x)\end{array}\right.\right.$.
Suy ra $\int_{0}^{3} x \cdot f^{\prime}(x) \mathrm{d} x=\left.x \cdot f(x)\right|_{0} ^{3}-\int_{0}^{3} f(x) \mathrm{d} x=3 f(3)-9=3.21-9=54$.
Vậy $I=6$.
Đáp án B.