How to evaluate $\int x \ln^n (x) dx$?

Question

How to evaluate $$\int x \ln^n (x) dx\,?$$

I tried to integrate through the parts. I used the formula, but I didn't know how to complete it.I found this solution, but I don't understand where that x comes from.

Answer 1

Let $u=\ln ^n(x)$; then we have $$ I_n(x)=\int x \ln^n(x)\,dx = \frac{1}{2}x^2\ln^n(x) -\frac{1}{2}\int x^2\cdot n \ln^{n-1}(x)\frac{1}{x} \,dx $$ $$ I_n(x) = \frac{1}{2}x^2\ln^n(x) -\frac{n}{2} I_{n-1}(x) $$If $n\in \mathbb{N}^+$, this can be continued until you get to $\int x\,dx$.

Answer 2

Another way

Let $x=e^t$ to make $$I_n=\int x \log^n(x)\,dx=\int t^n\, e^{2t}\,dt$$ Now make $t=-\frac u 2$ and you will face incomplete gamma functions.