Verified answer

[tex]\int x \ln(x) dx \\u = \ln(x)\quad du = \frac{dx}{x}\\dv = xdx \quad v = \frac{x^2}{2}\\\int x \ln(x) dx = \frac{x^2}{2}\ln(x) - \int \frac{xdx}{2} = \frac{x^2}{2}\ln(x) - \frac{x^2}{4} + C[/tex]