My question is strictly operative, if I have, for instance, two random variables $X$ and $Y$, $X$ is a $\mathcal{N}(m,\sigma^2)$ and $Y=e^{h(X-m)-1/2(h^2\sigma^2)}$.
$E[Ye^X]$ is $\int y e^x p(x) dx$, isn't it?
Sorry but I can't find an unambiguous answer, thanks in advance
That's about right. The convention is to use capital letters for the random variables themself, and lowercase for values they take in particular instances; such as the parameters in the integral.
$$\begin{align} \operatorname{\bf E}[Y{\bf e}^X] & = \int_{\bf \Omega} y\; {\bf e}^x \, {p}_{_X}(x) \operatorname{d} x\qquad:\quad y={\bf e}^{h(x-m)-1/2(h^2\sigma^2)} \\[1ex]& = \int_{\bf \Omega} \frac{{\bf e}^{h(x-m)-1/2(h^2\sigma^2) + x -(x-m)^2/2\sigma^2}}{\sigma\sqrt{2\pi}} \operatorname{d}x \end{align}$$