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Calculus Level 3

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

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 .

1 solution

Aditya Narayan Sharma
May 6, 2016

We have,

$\large \displaystyle\int_{0}^{1} Li_{2}(e^x) dx= \sum_{k=1}^{\infty} \frac{1}{k^2} \int_{0}^{1} e^{kx} dx$

$\large \displaystyle= \sum_{k=1}^{\infty} \frac{e^k-1}{k^3}$

$\large \displaystyle =Li_3(e)-\zeta(3)$ .

So, Answer is $\boxed{3+3+1=7}$