We define $E _ { 2 } := \left\{ ( p , q ) \in \left( \mathbf { N } ^ { * } \right) ^ { 2 } : p < q , p \mid q \right\}$ and $S _ { p , q } = \int _ { 0 } ^ { 1 } \dfrac { t ^ { q - 1 } } { 1 + t ^ { p } } d t$.
For all pairs $( p , q ) \in E _ { 2 }$, show that there exists a constant $\lambda := \lambda ( p , q )$ which one will determine, such that
$$S _ { p , q } = \frac { ( - 1 ) ^ { \lambda - 1 } } { p } \left( \ln ( 2 ) - \sum _ { k = 1 } ^ { \lambda - 1 } \frac { ( - 1 ) ^ { k - 1 } } { k } \right)$$