I am currently learning about the Cramér-Rao bound. According to the lecture, $\mathbb{E}_\vartheta [U_\vartheta(X_1)] = 0$, where $\vartheta$ is a parameter, $X_1$ is a random variable with density $f(x, \vartheta)$ and $U_\vartheta(X_1)$ is the score function $$U_\vartheta(X_1) := \frac{\partial \log f (X_1, \vartheta)}{\partial \vartheta}$$
We have 4 regularization constraints, but for this question I think we only need (R4):
(R4) $\int f(x, \vartheta) dx$ may be differentiated under the integral sign twice
I don't unterstand the first part of the proof:
From $\int_{-\infty}^{+ \infty} f(x, \vartheta) = 1$ follows with (R4):
$$\int_{-\infty}^{+ \infty} \frac{\partial f(x, \vartheta)}{\partial \vartheta} dx = 0$$
Why is that equal to 0?
(R4) gives that you can differentiate with respect to the parameters $\vartheta$, together with the others regularization constraints, we have:
\begin{align*} \int_{-\infty}^{+ \infty} f(x, \vartheta)\text{d}x &= 1\\ \implies\partial_{\vartheta} \Big(\int_{-\infty}^{+ \infty} f(x, \vartheta)\text{d}x\Big) &= \partial_{\vartheta} (1)\\ \implies\int_{-\infty}^{+ \infty} \frac{\partial f(x, \vartheta)}{\partial \vartheta} \text{d}x &= 0 \end{align*}