\displaystyle
\int_0^1 x^n \ln x dx = \lim_{a\to 0} \int_a^1 x^n \ln x dx =
\lim_{a\to 0} \left. \left[x^{n+1} \left( \frac{\ln x}{n+1} - \frac{1}{(n+1)^2} \right) \right] \right|_a^1 =