I am trying to evaluate following integral for all positive n: \begin{equation} \int_{0}^{1} x^n \cdot \ln(x)^n \,dx \end{equation} ,using partial integration, \begin{equation} \int u dv = uv - \int duv \\ u = \ln(x)^n,\quad\quad du = \frac{n\cdot \ln(x)^{n-1}}{x},\quad\quad dv=x^n,\quad\quad v=\frac{x^{n+1}}{n+1} \end{equation} I get the following:
\begin{equation} \left[\frac{\ln(x)^n \cdot x^{n+1}}{n+1}\right]_0^1-\int_{0}^{1}\frac{n\cdot \ln(x)^{n-1}}{x}\frac{x^{n+1}}{n+1}dx \end{equation}
Using L'Hospitals rule I evaluated the term with the square brackets to be $0$, so now I only have: \begin{equation} \frac{-n}{n+1}\int_{0}^{1}\ln(x)^{n-1}\cdot x^{n}dx \end{equation} which is a nice recursive Integral, which yields: \begin{equation} \frac{(-1)^{n-1}\cdot n!}{(n+1)^n} \end{equation}
But for my further calculations to work out I am supposed to get the following: \begin{equation} \frac{(-1)^{n-1}\cdot n!}{n^n} \end{equation} Where is my mistake?