Integrate a power of logarithm

Question

Is there some way, how to solve this problem? $$ \int \ln^n(x) dx \text{, where } n \in \mathbb{N} $$ I really don't know, what to do with $n$.

Answer 1

Use integration by part: $$\begin{align} \int \ln^n(x)dx=\int(x)'\ln^n(x)dx \\ =x\ln^{n}(x)+\int x (\ln^{n}(x))dx \\ =x\ln^{n}(x)+\int x \frac{1}{x}n \ln^{n-1}(x)dx\\ =x\ln^{n}(x)+n\int \ln^{n-1}(x)dx \end{align}$$

Answer 2

Hint: Give $n$ concrete values: $n=0$, $n=1$ and $n=2$. See what you get (integration by parts will be useful) and how these three cases relate to each other. Then attack the "general" $n$ and hope you get a relation between $I_n$ and $I_{n-1}$, where $I_n = \int \ln^n(x) dx$.