For which a and b is $\int_0^{1/2} r^{a+n-1}|\log(r)|^b dr<\infty$? The problem I am working on asks which real values of a and b make $|x|^a|\log|x||^b$ integrable over $\{x \in \mathbb{R}: |x| < 1/2\}$, but I reinterpret the question to asking which real values of a and b make $\int_0^{1/2} r^{a+n-1}|\log(r)|^b dr<\infty$. The trouble now is I'm not clear on how to check this inequality over all of $\mathbb{R}$. Actually, guessing and checking hasn't found me a single pair of values for which the integral isn't finite, which I'm sure would be far too easy and anyway I wouldn't know how to prove it.
I just need a hint on how to rigorously check the finiteness of this integral, or if this is the wrong approach to the problem, what the right approach might be.
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$\ds{\int_{0}^{1/2}r^{a + n - 1}\verts{\ln\pars{r}}^{b}\,\dd r < \infty:\
     {\large ?}}$

With $\ds{\quad r \equiv \expo{t}\quad\imp\quad t = \ln\pars{r}}$\begin{align}
&\color{#c00000}{\int_{0}^{1/2}r^{a + n - 1}\verts{\ln\pars{r}}^{b}\,\dd r}
=-\int_{\infty}^{2}r^{- a - n - 1}\ln^{b}\pars{r}\,\dd r
=\int_{\ln\pars{2}}^{\infty}\expo{-\pars{a + n + 1}t}t^{b}\pars{\expo{t}\,\dd t}
\\[3mm]&=\int_{\ln\pars{2}}^{\infty}t^{b}\expo{-\pars{a + n}t}\,\dd t
\quad\mbox{which converges when}\quad\pars{a + n} > 0\quad\mbox{and}\quad b > -1
\end{align}

\begin{align}
&\color{#00f}{\large\int_{0}^{1/2}r^{a + n - 1}\verts{\ln\pars{r}}^{b}\,\dd r}
={1 \over \pars{a + n}^{b + 1}}\int_{\pars{a + n}\ln\pars{2}}^{\infty}t^{b}
\expo{-t}\,\dd t
\\[3mm]&=\color{#00f}{\large%
{\Gamma\pars{b + 1,\pars{a + n}\ln\pars{2}} \over \pars{a + n}^{b + 1}}}\,,\qquad
\pars{a + n} > 0\,,\quad b > - 1
\end{align}
where $\ds{\Gamma\pars{s,x}}$ is the
Incomplete Gamma Function.
