For every pair of natural integers $( p , q )$ and for every $\varepsilon \in ] 0,1 [$, we denote
$$I _ { p , q } = \int _ { 0 } ^ { 1 } t ^ { p } ( \ln t ) ^ { q } \mathrm {~d} t \quad \text { and } \quad I _ { p , q } ^ { \varepsilon } = \int _ { \varepsilon } ^ { 1 } t ^ { p } ( \ln t ) ^ { q } \mathrm {~d} t$$
Deduce that we have $\forall p \in \mathbb { N } , \forall q \in \mathbb { N } ^ { * } , \quad I _ { p , q } = - \frac { q } { p + 1 } I _ { p , q - 1 }$.