Answer
$$\ln 4-\frac{3}{4}$$
Work Step by Step
Given $$\int_{1}^{2} x \ln x d x$$
Let
\begin{align*}
u&=\ln x\ \ \ \ \ \ \ \ dv= xdx\\
du&= \frac{ 1}{x} dx\ \ \ \ \ \ \ \ v=\frac{1}{2}x^2
\end{align*}
Then
\begin{align*}
\int_{1}^{2} x \ln x d x&= \frac{1}{2}x^2 \ln x\bigg|_{1}^{2} -\frac{1}{2} \int_{1}^{2} x d x \\
&= \frac{1}{2}x^2 \ln x\bigg|_{1}^{2} -\frac{1}{4} x^2\bigg|_{1}^{2} \\
&=2\ln 2-\frac{3}{4} \\
&= \ln 4-\frac{3}{4}
\end{align*}
