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Câu hỏi: Cho hàm số $y=f\left( x \right)$ liên tục trên $\mathbb{R}$. Gọi $F\left( x \right);G\left( x \right)$ là hai nguyên hàm của $f\left( x \right)$ trên $\mathbb{R}$ thỏa mãn $F\left( 2 \right)+2023.G\left( 0 \right)=5$ và $F\left( 0 \right)+2023G\left( 2 \right)=2$. Khi đó $\int\limits_{3}^{5}{f\left( 5-x \right)dx}$ bằng
A. $2023$.
B. $-\dfrac{3}{2022}$.
C. $3$.
D. $\dfrac{3}{2022}$.
A. $2023$.
B. $-\dfrac{3}{2022}$.
C. $3$.
D. $\dfrac{3}{2022}$.
Vì $F\left( x \right);G\left( x \right)$ là hai nguyên hàm của $f\left( x \right)$ trên $\mathbb{R}$ nên ta có:
$\int\limits_{0}^{2}{f\left( x \right)dx}=F\left( x \right)\left| _{0}^{2} \right.=F\left( 2 \right)-F\left( 0 \right)$ và $\int\limits_{0}^{2}{f\left( x \right)dx}=G\left( x \right)\left| _{0}^{2} \right.=G\left( 2 \right)-G\left( 0 \right)$
$\Rightarrow F\left( 2 \right)-F\left( 0 \right)=G\left( 2 \right)-G\left( 0 \right)$
Theo giả thiết ta có: $\left\{ \begin{aligned}
& F\left( 2 \right)+2023.G\left( 0 \right)=5 \\
& F\left( 0 \right)+2023G\left( 2 \right)=2 \\
\end{aligned} \right.$
Lấy vế trừ vế ta được: $\left[ F\left( 2 \right)-F\left( 0 \right) \right]-2023\left[ G\left( 2 \right)-G\left( 0 \right) \right]=3\Leftrightarrow F\left( 2 \right)-F\left( 0 \right)=-\dfrac{3}{2022}$
Xét $I=\int\limits_{3}^{5}{f\left( 5-x \right)dx}$. Đặt $t=5-x\Rightarrow dt=-dx$
$\Rightarrow I=\int\limits_{0}^{2}{f\left( t \right)dt}=\int\limits_{0}^{2}{f\left( x \right)dx}=F\left( 2 \right)-F\left( 0 \right)=-\dfrac{3}{2022}$
$\int\limits_{0}^{2}{f\left( x \right)dx}=F\left( x \right)\left| _{0}^{2} \right.=F\left( 2 \right)-F\left( 0 \right)$ và $\int\limits_{0}^{2}{f\left( x \right)dx}=G\left( x \right)\left| _{0}^{2} \right.=G\left( 2 \right)-G\left( 0 \right)$
$\Rightarrow F\left( 2 \right)-F\left( 0 \right)=G\left( 2 \right)-G\left( 0 \right)$
Theo giả thiết ta có: $\left\{ \begin{aligned}
& F\left( 2 \right)+2023.G\left( 0 \right)=5 \\
& F\left( 0 \right)+2023G\left( 2 \right)=2 \\
\end{aligned} \right.$
Lấy vế trừ vế ta được: $\left[ F\left( 2 \right)-F\left( 0 \right) \right]-2023\left[ G\left( 2 \right)-G\left( 0 \right) \right]=3\Leftrightarrow F\left( 2 \right)-F\left( 0 \right)=-\dfrac{3}{2022}$
Xét $I=\int\limits_{3}^{5}{f\left( 5-x \right)dx}$. Đặt $t=5-x\Rightarrow dt=-dx$
$\Rightarrow I=\int\limits_{0}^{2}{f\left( t \right)dt}=\int\limits_{0}^{2}{f\left( x \right)dx}=F\left( 2 \right)-F\left( 0 \right)=-\dfrac{3}{2022}$
Đáp án B.