∫ ( ln ( x ) ) 2 x d x = ∫ u 2 d u = u 2 + 1 2 + 1 d u = 1 3 u 3 + C u = ln ( x ) d u = 1 x d x = 1 3 ( ln ( x ) ) 3 + C {\displaystyle {\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}}dx\\&=\ {\frac {1}{3}}(\ln(x))^{3}+C\end{aligned}}}
![{\displaystyle {\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}}dx\\&=\ {\frac {1}{3}}(\ln(x))^{3}+C\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/14c67c7e2833b4725e660286233fe1fe30b4f5db)