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Câu hỏi: Cho hàm số $f\left( x \right)$ thỏa $2f\left( 1 \right)-f\left( 0 \right)=2$ và $\int\limits_{0}^{1}{\left( x+1 \right){f}'\left( x \right)\text{d}x}=10$. Tính $\int\limits_{0}^{1}{f\left( x \right)\text{d}x}$
A. $I=-8$.
B. $I=-12$.
C. $I=8$.
D. $I=1$.
A. $I=-8$.
B. $I=-12$.
C. $I=8$.
D. $I=1$.
Đặt $\left\{ \begin{aligned}
& u=x+1 \\
& dv={f}'\left( x \right) \\
\end{aligned} \right.\Rightarrow \left\{ \begin{aligned}
& du=dx \\
& v=f\left( x \right) \\
\end{aligned} \right.\Rightarrow $ $ \int\limits_{0}^{1}{\left( x+1 \right){f}'\left( x \right)\text{d}x}=\left( x+1 \right)f\left( x \right)|_{0}^{1}-\int\limits_{0}^{1}{f\left( x \right)\text{d}x}=10$.
$2f\left( 1 \right)-f\left( 0 \right)-\int\limits_{0}^{1}{f\left( x \right)\text{d}x}=10\Rightarrow \int\limits_{0}^{1}{f\left( x \right)\text{d}x}=-8$.
& u=x+1 \\
& dv={f}'\left( x \right) \\
\end{aligned} \right.\Rightarrow \left\{ \begin{aligned}
& du=dx \\
& v=f\left( x \right) \\
\end{aligned} \right.\Rightarrow $ $ \int\limits_{0}^{1}{\left( x+1 \right){f}'\left( x \right)\text{d}x}=\left( x+1 \right)f\left( x \right)|_{0}^{1}-\int\limits_{0}^{1}{f\left( x \right)\text{d}x}=10$.
$2f\left( 1 \right)-f\left( 0 \right)-\int\limits_{0}^{1}{f\left( x \right)\text{d}x}=10\Rightarrow \int\limits_{0}^{1}{f\left( x \right)\text{d}x}=-8$.
Đáp án A.