Multiple integral related to zeta function I am trying to calculate the following integral
$$\int_V \frac{d^d\vec{r}}{e^{x_1+...+x_d}-1},$$
where $V=[0,\infty)^d$ and $\vec{r}=(x_1,...,x_d)$. I know that the result should be related to the Riemann zeta function, but I do not see how to do it quickly and elementary (i.e. without the knowledge of all possible relations for zeta function). Any sugestion or hint ?

By Fubini's theorem
$$\int_{(0,+\infty)^d}\frac{x_1}{e^{x_1+\ldots+x_d}-1}\,d\mu_d = \int_{(0,+\infty)^{d-1}}\int_{0}^{+\infty}\frac{x_1}{e^{x_1+\ldots+x_d}-1}\,dx_1\,d\mu_{d-1} $$
and the last integral equals
$$ \int_{(0,+\infty)^{d-1}}\text{Li}_2\left(e^{-(x_2+\ldots+x_d)}\right)\,d\mu_{d-1}=\int_{(0,1)^{d-1}}\frac{\text{Li}_2(v_2\cdots v_d)}{v_2\cdots v_d}\,dv_2\cdot dv_d$$
or
$$ \int_{(0,1)^{d-2}}\frac{\text{Li}_3(v_3\cdots v_d)}{v_3\cdots v_d}dv_3\cdots dv_d = \int_{0}^{1}\frac{\text{Li}_d(w)}{w}\,dw=\text{Li}_{d+1}(1)=\color{red}{\zeta(d+1)}.$$
With the same technique we have:
$$\int_{(0,+\infty)^d}\frac{d\mu_d}{e^{x_1+\ldots+x_d}-1} = \text{Li}_d(1)=\color{red}{\zeta(d)}.$$
