Ta có: $(1+e)x=(1+e^x)x \Leftrightarrow \left[\begin{array}{l}x=0\\x=1\end{array}\right.$
Diện tích hình phẳng cần tìm là:
$S=\int\limits_0^1[(1+e)x-(1+e^x)x]dx$
$=\int\limits_0^1(ex-e^xx)dx$
$=\dfrac{ex^2}{2}\left|\begin{array}{l}1\\0\end{array}\right.-\int\limits_0^1e^xxdx$
$=\dfrac{e}{2}-\int\limits_0^1xd(e^x)$
$=\dfrac{e}{2}-xe^x\left|\begin{array}{l}1\\0\end{array}\right.+\int\limits_0^1 e^xdx$
$=\dfrac{-e}{2}+e^x\left|\begin{array}{l}1\\0\end{array}\right.$
$=\dfrac{e}{2}-1$