Cho hàm số f(x) có đạo hàm liên...

C

- Hướng dẫn giải

Đáp án C

$\begin{array}{l}\Rightarrow f\text{'}\left(x\right)+x{e}^x=0,\forall x\in \text{[}0;1\left]\right(do{\left(f\text{'}\left(x\right)+x{e}^x\right)}^2\ge 0,\forall x\in \text{[}0;1\left]\right)\\ \Rightarrow f\text{'}\left(x\right)=-x{e}^x\Rightarrow f\left(x\right)=(1-x)e^x+C\\ f\left(1\right)=0\Rightarrow f\left(x\right)=(1-x)e^x\\ \Rightarrow I=\underset{0}{\overset{1}{\int }}f\left(x\right)dx=\underset{0}{\overset{1}{\int }}(1-x)e^{x}dx=(2-x)e^x\left|\begin{array}{l}1\\ 0\end{array}\right.=e-2\end{array}$