quantum field theory - Supersymmetric Localization (Mirror Symmetry)

I'm reading Chapter 9 of Mirror Symmetry book. As you can see in eq. (9.30) his model for SUSY is

$$\begin{align} \delta_\epsilon X &=\epsilon^1\psi_1 + \epsilon^2\psi_2\\ \delta\psi_1 &= \epsilon^2\partial h\\ \delta\psi_2 &=-\epsilon^1\partial h. \end{align}\tag{9.30}$$

I understand that if $\partial h\neq 0$ it is possible to see that partition function vanishes, in particular, there is a total divergence:

$$ \int \mathrm{d}\hat{X}e^{\frac{1}{2}(\partial h(\hat{X}))^2}\frac{\partial^2 h(\hat{X})}{(\partial h(\hat{X}))^2}. \tag{*} $$

The implication that a total divergence yields vanishing integral, it is a bit strange, because, since every function that admit a primitive is a total derivative. What is the meaning of that implication? Then why $(*)$ vanishes? It seems to me that, in general, $$ \int \mathrm{d}(h')\frac{e^{\frac{1}{2}(h')^2}}{(h')^2}\neq 0. $$

Then, again, when he claims that if there is a $x_0$ such that $h'(x_0)=0$ we can use again the argument of the total divergence, why doesn't he keep in account that, removing a small neighborhood of $x_0$, give us some contribution at the border of that neighborhood (two points)?