Reference for closed form integral of $\int_0^1 dz\,z^n/(z-a)$ Is there a closed form (non-recursive) expression for the definite integral
$$\int_0^1 dz \frac{z^n}{z-a}, \qquad n\in\mathbb{Z}_+ \text{ and } a\notin (0,1)$$
for general $n$ and $a$ given in terms of logarithms?  Mathematica is able to give them for any given integer $n$ (I made a table for $n=\{0,\ldots, 5\}$), but I can't find in the literature how the incomplete Beta function is related to logarithms:

Anyone pointing me to the appropriate formulae in the NIST Handbook of Mathematical functions or in Gradshteyn and Ryzhik would be fantastic.
 A: Solution: You may write 
$$\begin{align}
\int_0^1  \frac{z^n}{z-a}dz&=\int_0^1 \frac{z^n-a^n}{z-a}dz+a^n\int_0^1 \frac{1}{z-a}dz\\ 
&=\int_0^1 \sum_{k=0}^{n-1}a^{n-1-k}z^kdz+a^n\int_0^1 \frac{1}{z-a}dz\\ 
&=\sum_{k=0}^{n-1}a^{n-k-1}\int_0^1 z^kdz+a^n\left. \log (z-a)\right|_0^1\\ 
&=\sum_{k=1}^n\frac{a^{n-k}}{k}+a^n \log \left(1-\frac1a\right)
\end{align}
$$
A: $\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
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$\ds{\int_{0}^{1}\dd z\,{z^{n} \over z - a}\,,\qquad n\in{\mathbb Z}_{+}\
     \mbox{and}\ a \not\in \pars{0,1}:\ {\large ?}}$.

Lets $\ds{{\cal I}_{n} \equiv \int_{0}^{1}{z^{n} \over z - a}\,\dd z}$ such that

\begin{align}
{\cal I}_{n}&=\int_{0}^{1}{z^{n - 1}\pars{z - a} + a z^{n- 1} \over z - a}\,\dd z
={1 \over n} + a{\cal I}_{n - 1}
\end{align}

Then,
  \begin{align}
{\cal I}_{n}&={1 \over n} + {a \over n - 1} + a^{2}\,{\cal I}_{n - 2}
={1 \over n} + {a \over n - 1} + {a^{2} \over n - 2} + a^{3}\,{\cal I}_{n - 3}
\\[5mm]&=\cdots=\sum_{k\ =\ 0}^{n - 1}{a^{k} \over n - k} + a^{n}\,{\cal I}_{0}
\quad\mbox{where}\quad
{\cal I}_{0}=\int_{0}^{1}{\dd z \over z - a}=\ln\pars{\verts{1 - {1 \over a}}}
\end{align}

$$\color{#66f}{\large%
\int_{0}^{1}{z^{n} \over z - a}\,\dd z}
=\color{#66f}{\large{\sum_{k\ =\ 0}^{n - 1}{a^{k} \over n - k} + a^{n}\,\ln\pars{\verts{1 - {1 \over a}}}}}\,,\qquad
n\in{\mathbb Z}_{+}\ \mbox{and}\ a \not\in \pars{0,1}
$$
