The problem is to find an integrable function that bounds $f_n(x) = \frac{n^p x^r \log x}{1 + n^2 x^2}$ where $r>0$, $p<\min \{2,1+r\}$ so we can calculate $$\lim_{n \to \infty} \int_0^1 \frac{n^p x^r\log x}{1 + n^2 x^2}dx = \int_0^1 \lim_{n \to \infty} \frac{n^p x^r\log x}{1 + n^2 x^2} dx = 0$$ using dominated convergence theorem.
I know how to solve this problem by taking derivatives with respect to $n$, but I have difficulty finding a solution without derivatives. I have tried $1+n^2x^2\ge 2nx$, but that does not look helpful.
Thank you!