Local Utility Functions
Locally, it is common to approximate annoying functions using polynomials. One widely-used approach is just a linear approximation Linear approximation. In Wikipedia. This can be invaluable, but it sucks as an approximation of a utility function, because utility functions are unique only up to linear transformation anyway Utility, so any linear approximation is just as good as any other (except in the trivial case where the slope is 0).
The quadratic approximation, however, can be much more useful, so let's take a quick look into it. Suppose our utility function is
$$ u(x) = c_0 + c_1 \cdot x + c_2 \cdot x^2 $$And suppose $x$ is a normal random variable. What is our expected utility? The answer is
$$ E[u(X)] = \int_{-\infty}^{\infty}{ \left( c_0 + c_1 \cdot x + c_2 \cdot x^2 \right) \frac{e^{\frac{{(x-\mu)}^2}{-2\sigma^2}}}{\sigma \sqrt{2\pi}} dx} $$which, we can simplify using properties of the normal distribution's moments Normal distribution. In Wikipedia to just
$$ E[u(X)] = c_0 + c_1 \cdot \mu + c_2 (\mu^2 + \sigma^2) $$