# Change in normal integrals

Calculus Level 5

$\large \displaystyle\int _{ 0 }^{ 1 }{ \ln { \left( 1-{ e }^{ \sqrt { x } } \right) \, dx } }$ The above integral is in the form: $a\left[\text{Li}_b(e^{-c})+\text{Li}_d(e^{-f}) -\zeta (g)+\frac{1}{h} \right]+i\pi,$

where $$a,b,c,d,f,g$$ and $$h$$ are positive integers. Find $$a+b+c+d+f+g+h-2$$.

Notations:

• $${ \text{Li} }_{ n }(a)$$ 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.

• $$i=\sqrt{-1}$$

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