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Câu hỏi: Giả sử hàm $f$ có đạo hàm cấp 2 trên $\mathbb{R}$ thỏa mãn $f\left( 1 \right)={f}'\left( 1 \right)=1$ và $f\left( 1-x \right)+{{x}^{2}}{{f}^{n}}\left( x \right)=2x,\forall x\in \mathbb{R}$. Tính tích phân $I=\int\limits_{0}^{1}{x{f}'\left( x \right)dx}$
A. $I=1$
B. $I=2$
C. $I=\dfrac{1}{3}$
D. $I=\dfrac{2}{3}$
A. $I=1$
B. $I=2$
C. $I=\dfrac{1}{3}$
D. $I=\dfrac{2}{3}$
Đặt $\left\{ \begin{aligned}
& u={f}'\left( x \right) \\
& dv=xdx \\
\end{aligned} \right.\Leftrightarrow \left\{ \begin{aligned}
& du={{f}'}'\left( x \right)dx \\
& v=\dfrac{{{x}^{2}}}{2} \\
\end{aligned} \right.\Rightarrow 2I=\left. {{x}^{2}}{f}'\left( x \right) \right|_{0}^{1}-\int\limits_{0}^{1}{{{x}^{2}}{{f}'}'\left( x \right)dx}$
$\Leftrightarrow 2I=1-\int\limits_{0}^{1}{{{x}^{2}}{{f}'}'\left( x \right)dx}$ suy ra
Ta có $f\left( 1-x \right)+{{x}^{2}}{{f}'}'\left( x \right)=2x\Leftrightarrow \int\limits_{0}^{1}{f\left( 1-x \right)dx+\int\limits_{0}^{1}{{{x}^{2}}{{f}'}'\left( x \right)dx=\int\limits_{0}^{1}{2xdx}}}$
$\Leftrightarrow \int\limits_{0}^{1}{f\left( x \right)dx+1-2I=\left. {{x}^{2}} \right|_{0}^{1}}=1\Rightarrow $ (1)
Đặt $\left\{ \begin{aligned}
& u=f\left( x \right) \\
& dv=dx \\
\end{aligned} \right.\Leftrightarrow \left\{ \begin{aligned}
& du={f}'\left( x \right)dx \\
& v=x \\
\end{aligned} \right.\Rightarrow $ (2)
Từ (1), (2) suy ra $2I=1-I\to I=\dfrac{1}{3}$.
& u={f}'\left( x \right) \\
& dv=xdx \\
\end{aligned} \right.\Leftrightarrow \left\{ \begin{aligned}
& du={{f}'}'\left( x \right)dx \\
& v=\dfrac{{{x}^{2}}}{2} \\
\end{aligned} \right.\Rightarrow 2I=\left. {{x}^{2}}{f}'\left( x \right) \right|_{0}^{1}-\int\limits_{0}^{1}{{{x}^{2}}{{f}'}'\left( x \right)dx}$
$\Leftrightarrow 2I=1-\int\limits_{0}^{1}{{{x}^{2}}{{f}'}'\left( x \right)dx}$ suy ra
Ta có $f\left( 1-x \right)+{{x}^{2}}{{f}'}'\left( x \right)=2x\Leftrightarrow \int\limits_{0}^{1}{f\left( 1-x \right)dx+\int\limits_{0}^{1}{{{x}^{2}}{{f}'}'\left( x \right)dx=\int\limits_{0}^{1}{2xdx}}}$
$\Leftrightarrow \int\limits_{0}^{1}{f\left( x \right)dx+1-2I=\left. {{x}^{2}} \right|_{0}^{1}}=1\Rightarrow $ (1)
Đặt $\left\{ \begin{aligned}
& u=f\left( x \right) \\
& dv=dx \\
\end{aligned} \right.\Leftrightarrow \left\{ \begin{aligned}
& du={f}'\left( x \right)dx \\
& v=x \\
\end{aligned} \right.\Rightarrow $ (2)
Từ (1), (2) suy ra $2I=1-I\to I=\dfrac{1}{3}$.
Đáp án C.