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[tex]\large\boxed{\begin{array}{l}\displaystyle\sf\int \sqrt{x}\,\ln(x)\,dx\\\sf u=\ln(x)\longrightarrow du=\dfrac{1}{x}\,dx\\\\\sf dv=\sqrt{x}\longrightarrow v=\dfrac{2}{3}x^{\frac{3}{2}}\\\\\displaystyle\sf\int\sqrt{x}\,\ln(x)\,dx=\dfrac{2}{3}x^{\frac{3}{2}}\ln(x)-\int\dfrac{2}{3}x^{\frac{3}{2}}\cdot\dfrac{1}{x}\,dx\\\\\displaystyle\sf\int\sqrt{x}\,\ln(x)\,dx=\dfrac{2}{3}x^{\frac{3}{2}}-\dfrac{2}{3}\int x^{\frac{1}{2}}\,dx\end{array}}[/tex]
[tex]\Large\boxed{\begin{array}{l}\displaystyle\sf\int\sqrt{x}\,\ln(x)\,dx=\dfrac{2}{3}x^{\frac{3}{2}}\ln(x)-\dfrac{2}{3}\cdot\dfrac{2}{3}x^{\frac{3}{2}}+k\\\boxed{\boxed{\boxed{\boxed{\displaystyle\sf\int\sqrt{x}\,\ln(x)\,dx=\dfrac{2}{3}x^{\frac{3}{2}}\ln(x)-\dfrac{4}{9}x^{\frac{3}{2}}+k}}}}\end{array}}[/tex]