How do I Calculate, if possible, in terms of well-known constants the integral :


$\int\_{0}^{1}x^{k}\psi(x)dx$ , where $k\geq 3$ is an integer ?


note: $\psi(x)$ is digamma function.


Any help would be greatly appreciated.



How do I Calculate :$\int_{0}^{1}x^{k}\psi(x)dx$ where $k\geq 3$ is an integer?
This integral has been observed by Donal F. Connor in 2010 (you can find the link [here](http://arxiv.org/abs/1005.3469), pg. 94). As far as I know, he found a closed form for the odd case, but I believe the even case is somewhere in that document (don't quote me on that yet). To solve it, as Feldmann Denis notes, use integration by parts.


According to Dr. Connor, we have $$ \int\_0^1 x^{2n+1}\psi{(x)} \;dx = 
\sum\_{k=0}^{2n} \left(
 \begin{array}{c}
 2n+1 \\
 k
 \end{array}
 \right) \big(H\_k\zeta{(-k)} + (-1)^{k+1}\zeta'{(-k)}\big)$$


where $H\_k$ is the harmonic series, and $\zeta$ is the Riemann zeta function.


