7.1 Integration By Parts/47

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

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

Failed to parse (unknown function "\begin{align}"): {\displaystyle \begin{align} \int_{}^{} \left(\ln(x)^{n}\right)dx = x \ln(x)^{n} - \int_{}^{} \left((x \frac{n \ln(x)^{n-1}}{x}) \right)dx \\[2ex] &= x \ln(x)^{n} - \int_{}^{} \left(n \ln(x)^{n-1}} \right)dx \end{align} }