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Câu hỏi: Cho hàm số $f(x)$ thỏa mãn $f'(x)=x.{{e}^{x}}$ và $f(0)=2$. Tính $f(1)$
A. $f(1)=8-2e$
B. $f(1)=5-e$
C. $f(1)=e$
D. $f(1)=3$
A. $f(1)=8-2e$
B. $f(1)=5-e$
C. $f(1)=e$
D. $f(1)=3$
Ta có: $\int\limits_{0}^{1}{{f}'\left( x \right)}dx=f\left( 1 \right)-f\left( 0 \right)\Leftrightarrow f\left( 1 \right)=f\left( 0 \right)+\int\limits_{0}^{1}{x{{e}^{x}}dx}$
Ta có: $\int\limits_{0}^{1}{x{{e}^{x}}dx}=\int\limits_{0}^{1}{xd\left( {{e}^{x}} \right)}=x{{e}^{2}}\left| \begin{aligned}
& 1 \\
& 0 \\
\end{aligned} \right.-\int\limits_{0}^{1}{{{e}^{x}}dx}=e-{{e}^{x}}\left| \begin{aligned}
& 1 \\
& 0 \\
\end{aligned} \right.=1\Rightarrow f\left( 1 \right)=3$.
Ta có: $\int\limits_{0}^{1}{x{{e}^{x}}dx}=\int\limits_{0}^{1}{xd\left( {{e}^{x}} \right)}=x{{e}^{2}}\left| \begin{aligned}
& 1 \\
& 0 \\
\end{aligned} \right.-\int\limits_{0}^{1}{{{e}^{x}}dx}=e-{{e}^{x}}\left| \begin{aligned}
& 1 \\
& 0 \\
\end{aligned} \right.=1\Rightarrow f\left( 1 \right)=3$.
Đáp án D.