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Calculus Level 2

\[\large \int _{ 0 }^{ 1 }{\text{Li}_2(e^x) \, dx } \]

If the value of the integral above is equal to \[\text{Li}_a(be)-\zeta(c), \] where \(a,b\) and \(c\) are positive integers, find \(a+b+c\).

Notation:

  • \({ \text{Li} }_{ n }(a) \) denotes the polylogarithm function, \({ \text{Li} }_{ n }(a)=\displaystyle\sum _{ k=1 }^{ \infty }{ \frac { { a }^{ k } }{ { k }^{ n } } }. \)
  • \(\zeta(\cdot) \) denotes the Riemann zeta function.
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