Ta có: $x\ln x=0 \Leftrightarrow \left[\begin{array}{l}x=0\\x=1\end{array}\right.$
Diện tích hình phẳng cần tìm là:
$S=\int\limits_1^e|x\ln x|dx$
$=\int\limits_1^e x\ln xdx$
$=\int\limits_1^e\ln xd(\dfrac{x^2}{2})$
$=\dfrac{x^2\ln x}{2}\left|\begin{array}{l}e\\1\end{array}\right.-\int\limits_1^e\dfrac{x^2}{2}d(\ln x)$
$=\dfrac{e^2}{2}-\int\limits_1^e\dfrac{x}{2}dx$
$=\dfrac{e^2}{2}-\dfrac{x^2}{4}\left|\begin{array}{l}e\\1\end{array}\right.$
$=\dfrac{e^2+1}{4}$