Let $n$ be a fixed integer. Find a reduction formula for $$\int x^n\ln^k(x)\,dx$$ that relates $\int x^n\ln^k(x)\,dx$ to $\int x^n\ln^{k-1}(x)\,dx$.
What is this question asking for?
Am I just supposed to take the integral of it and leave it in terms of $k$ and $n$?
By parts:
$$u=\log^kx\;,\;\;u'=\frac{k\log^{k-1}x}{x}\\v'=x^n\;,\;\;v=\frac{x^{n+1}}{n+1}$$
and from here
$$I_{n,k}:=\int x^n\log^kxdx=\frac{x^{n+1}\log^kx}{n+1}-\frac{k}{n+1}I_{n,(k-1)}$$