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Câu hỏi: Biết $\int\limits_{0}^{1}{\left( 1-x \right){f}'\left( x \right)dx=2}$ và $f\left( 0 \right)=3$. Khi đó $\int\limits_{0}^{1}{f\left( x \right)dx}$ bằng
A. $-5$.
B. 1.
C. $-1$.
D. 5.
A. $-5$.
B. 1.
C. $-1$.
D. 5.
Đặt $\left\{ \begin{aligned}
& u=1-x \\
& dv=f'\left( x \right)dx \\
\end{aligned} \right.\Rightarrow \left\{ \begin{aligned}
& du=-dx \\
& v=f\left( x \right) \\
\end{aligned} \right.$
$\begin{aligned}
& \Rightarrow 2=\left( 1-x \right)f\left( x \right)\left| _{0}^{1} \right.+\int\limits_{0}^{1}{f\left( x \right)\text{d}x} \\
& \Leftrightarrow 2=-f\left( 0 \right)+\int\limits_{0}^{1}{f\left( x \right)\text{d}x} \\
& \Leftrightarrow 2=-3+\int\limits_{0}^{1}{f\left( x \right)\text{d}x} \\
& \Leftrightarrow \int\limits_{0}^{1}{f\left( x \right)\text{d}x}=5 \\
\end{aligned}$
& u=1-x \\
& dv=f'\left( x \right)dx \\
\end{aligned} \right.\Rightarrow \left\{ \begin{aligned}
& du=-dx \\
& v=f\left( x \right) \\
\end{aligned} \right.$
$\begin{aligned}
& \Rightarrow 2=\left( 1-x \right)f\left( x \right)\left| _{0}^{1} \right.+\int\limits_{0}^{1}{f\left( x \right)\text{d}x} \\
& \Leftrightarrow 2=-f\left( 0 \right)+\int\limits_{0}^{1}{f\left( x \right)\text{d}x} \\
& \Leftrightarrow 2=-3+\int\limits_{0}^{1}{f\left( x \right)\text{d}x} \\
& \Leftrightarrow \int\limits_{0}^{1}{f\left( x \right)\text{d}x}=5 \\
\end{aligned}$
Đáp án D.
