Power formulae without phasors

I wanted to derive the formulae for active, reactive and apparent power.
While I understand the concept of phasors, I think I would find an analytical proof that doesn’t involve phasors, more intuitive. I recently came across an article that explained how active power is basically the time average of the product of instantaneous voltage and current through the load. I could make sense of it.

I was wondering if there are similar derivations for reactive and apparent powers as well, that do not employ phasors.

What I thought of:

The derivation of active power led me to think that I could perhaps derive the other formulae too, using integrals.

I had the following equations with me:

$$begin{align}
I(t) &= I_m sin(omega t + phi)
V(t) &= V_m sin(omega t)
end{align}$$

When I decomposed $I(t)$ in terms of $sin(omega t)$ and $cos(omega t)$, I immediately saw that while finding the average of $V(t)I(t)$ over a time period, the term with $cos(omega t)$ would not contribute.

So, the term $I_m sin(phi)cos(omega t)$ would have something to do with the reactive power. However, I still am unable to logically arrive at the pre-established formula for reactive power: $dfrac{V_m I_m sin(phi)}{2}$.

I was unable to make a headway with the derivation of apparent power, either.

Any help or pointers would be greatly appreciated.