1. Ta có:
$\int\limits_1^e (x\ln x)^2dx$
$=\dfrac{1}{3}\int\limits_1^e(\ln x)^2d(x^3)$
$=\dfrac{x^3(\ln x)^2}{3} \left|\begin{array}{l}e\\1\end{array}\right.-\dfrac{1}{3}\int\limits_1^ex^3d((\ln x)^2)$
$=\dfrac{e^3}{3}-\dfrac{2}{3}\int\limits_1^ex^2\ln xdx$
$=\dfrac{e^3}{3}-\dfrac{2}{9}\int\limits_1^e\ln xd(x^3)$
$=\dfrac{e^3}{3}-\dfrac{2x^3\ln x}{9}\left|\begin{array}{l}e\\1\end{array}\right.+\dfrac{2}{9}\int\limits_1^ex^3d(\ln x)$
$=\dfrac{e^3}{3}-\dfrac{2e^3}{9}+\dfrac{2}{9}\int\limits_1^ex^2dx$
$=\dfrac{e^3}{9}+\dfrac{2x^3}{27}\left|\begin{array}{l}e\\1\end{array}\right.$
$=\dfrac{5e^3-2}{27}$