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Câu hỏi: Cho hàm số $y=f\left( x \right)$ có đạo hàm ${f}'\left( x \right)$ liên tục trên $\left[ 0;2 \right]$. Khi đó $\int\limits_{0}^{2}{x.{f}'\left( x \right)dx}$ bằng
A. $f\left( x \right)\left| _{\begin{smallmatrix}
\\
0
\end{smallmatrix}}^{\begin{smallmatrix}
2 \\
\end{smallmatrix}}-\int\limits_{0}^{2}{f\left( x \right)dx.} \right. $
B. $ x.f\left( x \right)\left| _{\begin{smallmatrix}
\\
0
\end{smallmatrix}}^{\begin{smallmatrix}
2 \\
\end{smallmatrix}}-\int\limits_{0}^{2}{f\left( x \right)dx.} \right.$
C. $x\left| _{\begin{smallmatrix}
\\
0
\end{smallmatrix}}^{\begin{smallmatrix}
2 \\
\end{smallmatrix}} \right.-\int\limits_{0}^{2}{f\left( x \right)dx.} $
D. $ x.f\left( x \right)\left| _{\begin{smallmatrix}
A. $f\left( x \right)\left| _{\begin{smallmatrix}
\\
0
\end{smallmatrix}}^{\begin{smallmatrix}
2 \\
\end{smallmatrix}}-\int\limits_{0}^{2}{f\left( x \right)dx.} \right. $
B. $ x.f\left( x \right)\left| _{\begin{smallmatrix}
\\
0
\end{smallmatrix}}^{\begin{smallmatrix}
2 \\
\end{smallmatrix}}-\int\limits_{0}^{2}{f\left( x \right)dx.} \right.$
C. $x\left| _{\begin{smallmatrix}
\\
0
\end{smallmatrix}}^{\begin{smallmatrix}
2 \\
\end{smallmatrix}} \right.-\int\limits_{0}^{2}{f\left( x \right)dx.} $
D. $ x.f\left( x \right)\left| _{\begin{smallmatrix}
Đặt $u=x\Rightarrow du=dx,d\upsilon ={f}'\left( x \right)dx$ nên chọn $\upsilon =\int{{f}'\left( x \right)dx=f\left( x \right)}$
Ta có $\int\limits_{0}^{2}{x.{f}'\left( x \right)dx=x.f\left( x \right)\left| _{\begin{smallmatrix}
\\
0
\end{smallmatrix}}^{\begin{smallmatrix}
2 \\
\end{smallmatrix}} \right.-\int\limits_{0}^{2}{f\left( x \right)dx}.}$
Ta có $\int\limits_{0}^{2}{x.{f}'\left( x \right)dx=x.f\left( x \right)\left| _{\begin{smallmatrix}
\\
0
\end{smallmatrix}}^{\begin{smallmatrix}
2 \\
\end{smallmatrix}} \right.-\int\limits_{0}^{2}{f\left( x \right)dx}.}$
Đáp án B.