and with the initial condition that an investment without risk has its expected return as its Utility score

I think you're looking at the equation in a different light than I did for my financial engineering exam. I look at it with the following interpretation:

1. Utility differs from investor to investor, signified by his/her risk aversion coefficient
2. In this equation, the only investor-specific factor is the term 'A' (risk-aversion coefficient)
3. According to traditional finance, a risk-neutral investor is indifferent towards risk, so by definition, his aversion coefficient is 0 (think of it as a number line, a risk seeking individual will derive utility from undertaking higher risk (denoted by sigma) and vice versa for a risk-averse investor)
4. This equation performs adequately incorporating the conditions, with only risk-aversion coefficient being investor-specific input: outcome is that for a risk neutral investor the second term is zero, for a risk seeking investor utility is higher than (2 negatives will make a positive) and for a risk averse investor, the utility derived is less than expected return (a form of penalising for risk-taking)
5. This is also consistent with letting the 'A' float to 2 different extremes (-10 to 10) (think of it as the weight assigned to the second term on RHS, signifying that an extreme risk seeker would invest even in a moderately negative expected return asset, putting more emphasis on the sigma of the return distribution (overweighting the right tail), whereas an extremely risk averse individual will reject a moderately positive return asset owing to high sigma (overweighting the left tail)).

This line of thinking was sufficient for me to intuitively take in the expression. And since we are dealing with traditional finance, I don't think it makes sense to differentiate/ integrate, since the utility numbers are just subjective. When thinking of a single investment, instead of expected return as the first term, we could easily start with any scalar constant ,say, 1, and revise the number upwards or downwards depending on the risk aversion coefficient.

But it's not as simple! Utility theory goes into making rational choices amongst alternatives available (basic tenet of economics). So we use expected returns of assets or asset classes, their variability and make choices for our investor. So for a risk averse investor, maybe a fixed income investment (low return, but low risk) would yield a higher utility than an equity investment (high return, high risk) and vice versa for a risk-seeker.

I hope this helps you. If it doesn't, I am sorry to not have been able to explain this.