  We investigate multivariate integration for a space of infinitely times
differentiable functions $\mathcal{F}_{s, \boldsymbol{u}} := \{f \in C^\infty
[0,1]^s \mid \| f \|_{\mathcal{F}_{s, \boldsymbol{u}}} < \infty \}$, where $\|
f \|_{\mathcal{F}_{s, \boldsymbol{u}}} := \sup_{\boldsymbol{\alpha} =
(\alpha_1, \dots, \alpha_s) \in \mathbb{N}_0^s}
\|f^{(\boldsymbol{\alpha})}\|_{L^1}/\prod_{j=1}^s u_j^{\alpha_j}$,
$f^{(\boldsymbol{\alpha})} := \frac{\partial^{|\boldsymbol{\alpha}|}}{\partial
x_1^{\alpha_1} \cdots \partial x_s^{\alpha_s}}f$ and $\boldsymbol{u} =
\{u_j\}_{j \geq 1}$ is a sequence of positive decreasing weights. Let $e(n,s)$
be the minimal worst-case error of all algorithms that use $n$ function values
in the $s$-variate case. We prove that for any $\boldsymbol{u}$ and $s$
considered $e(n,s) \leq C(s) \exp(-c(s)(\log{n})^2)$ holds for all $n$, where
$C(s)$ and $c(s)$ are constants which may depend on $s$. Further we show that
if the weights $\boldsymbol{u}$ decay sufficiently fast then there exist some
$1 < p < 2$ and absolute constants $C$ and $c$ such that $e(n,s) \leq C
\exp(-c(\log{n})^p)$ holds for all $s$ and $n$. These bounds are attained by
quasi-Monte Carlo integration using digital nets. These convergence and
tractability results come from those for the Walsh space into which
$\mathcal{F}_{s, \boldsymbol{u}}$ is embedded.
