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Câu hỏi: Cho hàm số $f(x)$ liên tục trên $\mathbb{R}$ và thỏa mãn $\int\limits_{0}^{3}{xf(x)\text{d}x}=2$. Tích phân $\int\limits_{0}^{1}{xf(3x)\text{d}x}$ bằng
A. $\dfrac{2}{3}$.
B. $18$
C. $\dfrac{2}{9}$.
D. $6$.
A. $\dfrac{2}{3}$.
B. $18$
C. $\dfrac{2}{9}$.
D. $6$.
Cách 1:
Đặt $t=3x\Rightarrow \text{d}t=3\text{d}x$
Ta có: $\int\limits_{0}^{1}{xf(3x)\text{d}x}=\int\limits_{0}^{3}{\dfrac{t}{3}f(t)\dfrac{\text{d}t}{3}}=\dfrac{1}{9}\int\limits_{0}^{3}{tf(t)\text{d}}t=\dfrac{2}{9}$
Cách 2:
$\int\limits_{0}^{1}{xf(3x)\text{d}x}=\dfrac{1}{9}\int\limits_{0}^{3}{(3x)f(3x)\text{d(3}x)}=\dfrac{2}{9}$
Đặt $t=3x\Rightarrow \text{d}t=3\text{d}x$
| $x$ | 0 1 |
| $t$ | 0 3 |
Cách 2:
$\int\limits_{0}^{1}{xf(3x)\text{d}x}=\dfrac{1}{9}\int\limits_{0}^{3}{(3x)f(3x)\text{d(3}x)}=\dfrac{2}{9}$
Đáp án C.