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Câu hỏi: Cho $f\left( x \right)$ là hàm số liên tục trên $\mathbb{R}$ thỏa mãn $\int\limits_{0}^{1}{f\left( x \right)\text{d}x=4}$ và $\int\limits_{0}^{1}{f\left( 3x \right)\text{d}x=6}$. Tích phân $\int\limits_{1}^{3}{f\left( x \right)}\text{d}x$ bằng
A. 10.
B. 2.
C. 12.
D. 14.
A. 10.
B. 2.
C. 12.
D. 14.
Đặt $I=\int\limits_{0}^{1}{f\left( 3x \right)\text{d}x=6}$.
Đặt $t=3x\Rightarrow \text{d}t=3\text{d}x$. Đổi cận $\left\{ \begin{aligned}
& x=0\Rightarrow t=0 \\
& x=1\Rightarrow t=3 \\
\end{aligned} \right.$.
$I=\int\limits_{0}^{3}{f\left( t \right)\frac{\text{d}t}{3}}=\frac{1}{3}\int\limits_{0}^{3}{f\left( x \right)\text{d}x}=\frac{1}{3}\left( \int\limits_{0}^{1}{f\left( x \right)\text{d}x+\int\limits_{1}^{3}{f\left( x \right)\text{d}x}} \right)=\frac{1}{3}\left( 4+\int\limits_{1}^{3}{f\left( x \right)dx} \right)$
$\Leftrightarrow 6=\frac{1}{3}\left( 4+\int\limits_{1}^{3}{f\left( x \right)\text{d}x} \right)\Leftrightarrow \int\limits_{1}^{3}{f\left( x \right)\text{d}x}=14$.
Đặt $t=3x\Rightarrow \text{d}t=3\text{d}x$. Đổi cận $\left\{ \begin{aligned}
& x=0\Rightarrow t=0 \\
& x=1\Rightarrow t=3 \\
\end{aligned} \right.$.
$I=\int\limits_{0}^{3}{f\left( t \right)\frac{\text{d}t}{3}}=\frac{1}{3}\int\limits_{0}^{3}{f\left( x \right)\text{d}x}=\frac{1}{3}\left( \int\limits_{0}^{1}{f\left( x \right)\text{d}x+\int\limits_{1}^{3}{f\left( x \right)\text{d}x}} \right)=\frac{1}{3}\left( 4+\int\limits_{1}^{3}{f\left( x \right)dx} \right)$
$\Leftrightarrow 6=\frac{1}{3}\left( 4+\int\limits_{1}^{3}{f\left( x \right)\text{d}x} \right)\Leftrightarrow \int\limits_{1}^{3}{f\left( x \right)\text{d}x}=14$.
Đáp án D.