du=\frac{1}{x}\\dv=xdx=>v=\frac{x^2}{2}\\\\ \int\limits^2_1 {x*ln(x)} \, dx=\frac{x^2}{2}lnx|^2_1- \int\limits^2_1 {\frac{x}{2}} \, dx=\frac{x^2}{2}lnx|^2_1-\frac{x^2}{4}|^2_1=2ln2-1+\frac{1}{4}=\\=2ln2-\frac{3}{4}" alt=" \int\limits^2_1 {x*ln(x)} \, dx\\\\ \int\limits^b_a {u} \, dv=uv|^b_a- \int\limits^b_a {v} \, du\\\\u=lnx=>du=\frac{1}{x}\\dv=xdx=>v=\frac{x^2}{2}\\\\ \int\limits^2_1 {x*ln(x)} \, dx=\frac{x^2}{2}lnx|^2_1- \int\limits^2_1 {\frac{x}{2}} \, dx=\frac{x^2}{2}lnx|^2_1-\frac{x^2}{4}|^2_1=2ln2-1+\frac{1}{4}=\\=2ln2-\frac{3}{4}" align="absmiddle" class="latex-formula">