# Expectation of some function with exponential form under Gaussian distribution

Suppose we have a real-valued function $g \colon \mathbb{R} \to \mathbb{R}$ (here, assume that we do not know the closed-form of $g$), and define its composition function $f$ by $f(x) := \exp (g(x))$.

Let $\mu$ be the standard normal distribution.

Is it possible to get an analytical form of the integration $\int f(x) \mu(\mathrm{d}x)$? If so, how to reach that?

• I guess there are not many functions $g$ that you are able to obtain an analytical expression for the integral, perhaps except the trivial case of linear and some quadratic functions. – BGM May 10 '17 at 8:18
• Yeah, you're right. I've got the case of linear functions $g(x) := c_0 + c_1 x$, and the quadratic case is quite similar to linear case. Thus, I am just curious about the general case for $g(x)$. Thanks @BGM – Paradiesvogel May 10 '17 at 22:59