I have a random variable, $X$, which happens to be lognormally distributed, or $\ln(X) \sim \mathcal{N}(\mu, \sigma^2)$, and I know the parameters of the lognormal distribution.
I wish to calculate the expectation of a some payoff that is solely a function of $X$, say $P(X)$ and so resolve to calculate $\int P(X) \cdot D(X) dx$, where $D(X)$ is the probably distribution function $X$; the definition for $D(X)$ is obviously known, but much more complex than the PDF for a normal distribution, and I could attempt to brute force the integral from here but I was wondering if there is a better way.
I was hoping to simply calculate $E(Y)$, where $Y = \ln(X)$, by the same formula as above $\int P(Y) \cdot D(Y) dy$, but here $D(Y)$ would be the normal distribution PDF instead of the lognormal distribution PDF. If I know the expected value of $Y$, my guess was I could calculate the expected value of $X$ by simply taking the exponential $\exp{Y}$, but I feel like this won't work. Can someone explain why this won't work and perhaps if there is a better solution?