Tetration of an integral?

After integrating $\Iota$ , we can see that $\Iota = \cfrac{1}{e}$

Now we can solve for $a$ :

$a = \lim\limits_{s\to\infty} {^{s}\Iota} = \Iota^{\Iota^{\Iota^{\dots}}} = \left(\frac{1}{e}\right)^{\left(\frac{1}{e}\right)^{\left(\frac{1}{e}\right)^{\dots}}}$

$\Longrightarrow a = \left(\frac{1}{e}\right)^{a} \Longrightarrow ae^{a}=1$ We can see that $ae^a = 1$ and from here we can conclude that

$\cfrac{1}{2} + \int_{0}^{1} ae^a+2\tau \: d\tau = \cfrac{1}{2} + \int_{0}^{1}1+2\tau \: d\tau = \cfrac{1}{2} + \left( \tau + \tau^2 \bigg|_{0}^{1} \right)$

$= \cfrac{1}{2} + \left( 2 - 0 \right) = \boxed{2.5}$