I have probability density function of the log normal distribution $f_X(x, \mu, \sigma^2)$, where $x$ is the random variable, $\mu$ is the location and $\sigma^2$ is the scale parameters of the distribution. For various sets of reasons I have realised that in order to solve a problem that I have, I might need to calculate the expectation of the PDF with respect to $x$. So this is not a $E(x)$, but it is $E(f_X(x, \mu, \sigma^2))$ that I need. In a way, I am saying that the PDF itself is a random variable. I would be able to calculate the expectation myself if I knew the distribution of the PDF of the log normal distribution. Unfortunately, I cannot find any information on this.
So, my question: is there a theorem, or any property, or anything else that would connect PDF of exponential family (log normal in my case) with the distribution of the PDF?