Step 1: ILATE priority:
By the ILATE rule, Logarithmic functions are integrated as \(u\) before Algebraic functions as \(dv\), confirming \(u=\log x,\ dv=x\,dx\).
Step 2: By-parts substitution:
With \(v=\dfrac{x^2}{2}\), \(du=\dfrac{dx}{x}\): \(\int x\log x\,dx=v\,u-\int v\,du=\dfrac{x^2}{2}\log x-\int\dfrac{x^2}{2}\cdot\dfrac{1}{x}dx\).
Step 3: Simplifying the leftover integrand:
\(\dfrac{x^2}{2}\cdot\dfrac1x=\dfrac{x}{2}\), whose integral is \(\dfrac{x^2}{4}\).
Final Answer:
\[ \boxed{\dfrac{x^2}{2}\log x-\dfrac{x^2}{4}+C} \]