Is the following integral identity true or not? Is the following statement true or not?$$\int_{-\infty}^\infty xf(x)\,dx = \left. {d\over{dt}} \int_{-\infty}^\infty e^{tx}f(x)\,dx\right|_{t = 0}$$

The statement is if Leibnitz's rule can be applied to $e^{tx}f(x)$ around $t=0$. Specifically in this case you need an integrable function $\theta$ such that $|xe^{tx}f(x)|\leq \theta(x)$ for $t$ near $0$. 
