\[\int_{0}^{a}x^{n}\psi\left(x\right)\,\mathrm{d}x=(-1)^{n-1}\zeta'\left(-n%
\right)+(-1)^{n}H_{n}\frac{B_{n+1}}{n+1}-\sum_{k=0}^{n}(-1)^{k}\genfrac{(}{)}{%
0.0pt}{}{n}{k}H_{k}\frac{B_{k+1}(a)}{k+1}a^{n-k}+\sum_{k=0}^{n}(-1)^{k}%
\genfrac{(}{)}{0.0pt}{}{n}{k}\zeta'\left(-k,a\right)a^{n-k},\]