∫ ( ln ( x ) n ) d x = x ( ln ( x ) n ) − n ∫ ( ln ( x ) n − 1 ) d x {\displaystyle \int _{}^{}\left(\ln(x)^{n}\right)dx=x\left(\ln(x)^{n}\right)-n\int _{}^{}\left(\ln(x)^{n-1}\right)dx}
u = ln ( x ) n d v = 1 d x d u = 1 d x v = x {\displaystyle {\begin{aligned}u&=\ln(x)^{n}\quad dv=1dx\\[2ex]du&=1dx\qquad v=x\\[2ex]\end{aligned}}}
∫ ( ln ( x ) n ) d x = x ln ( x ) n − ∫ ( ( x n ln ( x ) n − 1 x ) ) d x {\displaystyle {\begin{aligned}\int _{}^{}\left(\ln(x)^{n}\right)dx=x\ln(x)^{n}-\int _{}^{}\left((x{\frac {n\ln(x)^{n-1}}{x}})\right)dx\end{aligned}}}
![{\displaystyle {\begin{aligned}u&=\ln(x)^{n}\quad dv=1dx\\[2ex]du&=1dx\qquad v=x\\[2ex]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/11f9ca3ba4f0941cc2e612d695e552ba353ee8e3)
