7.1 Integration By Parts/47

$\int _{}^{}\left(\ln(x)^{n}\right)dx=x\left(\ln(x)^{n}\right)-n\int _{}^{}\left(\ln(x)^{n-1}\right)dx$

${\begin{aligned}u&=\ln(x)^{n}\quad dv=1dx\\[2ex]du&=1dx\qquad v=x\\[2ex]\end{aligned}}$

<math> \int_{}^{} \left(\ln(x)^{n}\right)dx = x\ln(x)^{n} - \int_{}^{} \left(x * \frac{n(\ln(x)^{n-1}}{x} \rigt)dx