[tex]\displaystyle
\int \sqrt x \ln x\, dx=(*)\\
t=\ln x,du=\sqrt x\, dx\\
dt=\dfrac{1}{x}\, dx, u=\dfrac{2}{3}x^{\tfrac{3}{2}}\\
(*)=\dfrac{2}{3}x^{\tfrac{3}{2}}\ln x-\int \dfrac{1}{x}\cdot\dfrac{2}{3}x^{\tfrac{3}{2}}\, dx=\\
\dfrac{2}{3}x^{\tfrac{3}{2}}\ln x-\dfrac{2}{3}\int \sqrt x\, dx=\\
\dfrac{2}{3}x^{\tfrac{3}{2}}\ln x-\dfrac{2}{3}\cdot\dfrac{2}{3}x^{\tfrac{3}{2}}=\\
\dfrac{2}{3}x^{\tfrac{3}{2}}\left(\ln x-\dfrac{2}{3}\right)+C[/tex]
