Evaluating $\int_{0}^{1} x^m \ln^\alpha(x) dx$ Honestly, I am asked to think about $$\int_{0}^{1} x^m \ln^\alpha(x) dx$$ And I applied all methods I know. I doubt if this integral makes sense either. If it is replicate, plz inform me to omit the question soon. Thanks.

$$
\begin{align}
\int_0^1x^m\,\log^\alpha(x)\,\mathrm{d}x
&=\int_{-\infty}^0u^\alpha\,e^{(m+1)u}\,\mathrm{d}u\\
&=(-1)^\alpha(m+1)^{-\alpha-1}\int_0^{\infty}t^\alpha e^{-t}\mathrm{d}t\\
&=(-1)^\alpha(m+1)^{-\alpha-1}\Gamma(\alpha+1)
\end{align}
$$
