5.5 The Substitution Rule/19

${\begin{aligned}&\int {\frac {\left(\ln(x)\right)^{2}}{x}}dx\ =\ \int u^{2}du\\[2ex]&=\ {\frac {u^{2+1}}{2+1}}du\ =\ {\frac {1}{3}}u^{3}+C\\[2ex]u=\ln(x)du={\frac {1}{x}}dxdu={\frac {dx}{x}}&=\ {\frac {1}{3}}(\ln(x))^{3}+C\end{aligned}}$