Prove ∫ ( 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}
Failed to parse (unknown function "\begin{align}"): {\displaystyle \begin{align} \int_{}^{} \left(\ln(x)^{n}\right)dx \[2ex] &u = \ln(x)^{n} \quad dv= 1dx \\[2ex] &du =1dx \quad v=x \\[2ex] \end{align} }
∫ ( ln ( x ) n ) d x = x ln ( x ) n − ∫ ( ( x n ln ( x ) n − 1 x ) ) d x = x ln ( x ) n − ∫ ( n ln ( x ) n − 1 ) d x = x ln ( x ) n − n ∫ ( ln ( x ) n − 1 ) 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\\[2ex]&=x\ln(x)^{n}-\int _{}^{}\left(n\ln(x)^{n-1}\right)dx\\[2ex]&=x\ln(x)^{n}-n\int _{}^{}\left(\ln(x)^{n-1}\right)dx\\[2ex]\end{aligned}}}
![{\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\\[2ex]&=x\ln(x)^{n}-\int _{}^{}\left(n\ln(x)^{n-1}\right)dx\\[2ex]&=x\ln(x)^{n}-n\int _{}^{}\left(\ln(x)^{n-1}\right)dx\\[2ex]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/19be1d5890fa5a4661b92c41d3728d2463eb1083)