Answer
$\displaystyle{I=\frac{16}{3}\ln2-\frac{29}{9}}\\$
Work Step by Step
$\displaystyle{x^2\ln x=4\ln x}\\
\displaystyle{x^2\ln x-4\ln x=0}\\
\displaystyle{\ln x\left(x^2-4\right)=0}\\
\displaystyle{\ln x=0 \qquad x^2-4=0}\\
\displaystyle{e^{\ln x}=e^0 \qquad x^2=4}\\
\displaystyle{x=e^0 \qquad x=\pm2\qquad x\gt0}\\
\displaystyle{x=1 \qquad x=2}\\
$
$\displaystyle{I=\int_{1}^{2}4\ln x-x^2\ln x\ dx}\\
\displaystyle{I=\int_{1}^{2}\ln x\left(x^2-4\right)\ dx}\\
$
$\displaystyle \left[\begin{array}{ll} u=\ln x & dv=4-x^2 \\ & \\ du=\frac{1}{x} & v=4x-\frac{1}{3}x^3 \end{array}\right]$ Integration by parts
$\displaystyle{I=\left[\ln x\times4x-\frac{1}{3}x^3\right]_1^2-\int_{1}^{2}\left(4x-\frac{1}{3}x^3\right)\left(\frac{1}{x}\right)\ dx}\\
\displaystyle{I=\frac{16}{3}\ln2-\int_{1}^{2}4-\frac{1}{3}x^2\ dx}\\
\displaystyle{I=\frac{16}{3}\ln2-\left[4x-\frac{1}{9}x^3\right]_{1}^{2}}\\
\displaystyle{I=\frac{16}{3}\ln2-\frac{29}{9}}\\$
