Solution
= Solution
Use <integration by parts> with $u=\ln x$ and $dv=x\,dx$. Since $du=dx/x$ and $v=x^2/2$,
$$
\int x\ln x\,dx=\frac{x^2}2\ln x-\frac12\int x\,dx
=\boxed{\frac{x^2}2\ln x-\frac{x^2}4+C},\qquad x>0.
$$
= Solution
Use <integration by parts> with $u=\ln x$ and $dv=x\,dx$. Since $du=dx/x$ and $v=x^2/2$,
$$
\int x\ln x\,dx=\frac{x^2}2\ln x-\frac12\int x\,dx
=\boxed{\frac{x^2}2\ln x-\frac{x^2}4+C},\qquad x>0.
$$