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Câu hỏi: Cho hàm số $y=f\left( x \right)$ liên trục trên $\left[ -4;0 \right]$ và có đồ thị như hình vẽ.
Biết diện tích các miền ${{S}_{1}}$, ${{S}_{2}}$, ${{S}_{3}}$ lần lượt là $\dfrac{124}{15},\dfrac{37}{60},\dfrac{53}{60}$. Giá trị của $I=\int\limits_{-6}^{-2}{x{f}'\left( x+2 \right)}\text{d}x$.
A. $-\dfrac{128}{15}$.
B. $\dfrac{293}{30}$.
C. $\dfrac{128}{15}$.
D. $-\dfrac{293}{30}$.
A. $-\dfrac{128}{15}$.
B. $\dfrac{293}{30}$.
C. $\dfrac{128}{15}$.
D. $-\dfrac{293}{30}$.
$I=\int\limits_{-6}^{-2}{x{f}'\left( x+2 \right)}\text{d}x=\int\limits_{-6}^{-2}{x\text{d}f\left( x+2 \right)}=\left. xf\left( x+2 \right) \right|_{-6}^{-2}-\int\limits_{-6}^{-2}{f\left( x+2 \right)\text{d}x}=-2f\left( -2 \right)+6f\left( -4 \right)-\int\limits_{-4}^{0}{f\left( x \right)\text{d}x}$
$=-\int\limits_{-4}^{-2}{f\left( x \right)\text{d}x}-\int\limits_{-2}^{-1}{f\left( x \right)\text{d}x}-\int\limits_{-1}^{0}{f\left( x \right)\text{d}x}=-\left( \dfrac{-124}{15} \right)-\dfrac{37}{60}-\left( \dfrac{-53}{60} \right)=\dfrac{128}{15}$.
$=-\int\limits_{-4}^{-2}{f\left( x \right)\text{d}x}-\int\limits_{-2}^{-1}{f\left( x \right)\text{d}x}-\int\limits_{-1}^{0}{f\left( x \right)\text{d}x}=-\left( \dfrac{-124}{15} \right)-\dfrac{37}{60}-\left( \dfrac{-53}{60} \right)=\dfrac{128}{15}$.
Đáp án C.