\(I=\int\limits^e_1x^2ln^2xdx\) (do \(\left(xlnx\right)^2>0\))
Đặt \(\left\{{}\begin{matrix}u=ln^2x\\dv=x^2dx\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}du=\dfrac{2lnx}{x}dx\\v=\dfrac{1}{3}x^3\end{matrix}\right.\)
\(I=\dfrac{1}{3}x^3ln^2x|^e_1-\dfrac{2}{3}\int\limits^e_1x^2lnxdx=\dfrac{1}{3}e^3-\dfrac{2}{3}I_1\)
Xét \(I_1=\int\limits^e_1x^2lnxdx\)
Đặt \(\left\{{}\begin{matrix}u=lnx\\dv=x^2dx\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}du=\dfrac{dx}{x}\\v=\dfrac{1}{3}x^3\end{matrix}\right.\)
\(I_1=\dfrac{1}{3}x^3lnx|^e_1-\dfrac{1}{3}\int\limits^e_1x^2dx=\dfrac{1}{3}e^3-\dfrac{1}{9}x^3|^e_1=\dfrac{2}{9}e^3+\dfrac{1}{9}\)
\(\Rightarrow I=\dfrac{1}{3}e^3-\dfrac{2}{3}\left(\dfrac{2}{9}e^3+\dfrac{1}{9}\right)=...\)
