Intensity in double slit interference

I’m a little confused about a small detail when finding the intensity over the screen, as some notes I’m using happen to calculate the same thing twice, with a slight difference the second time.

Each method does the following
Call the electric field at any point E so we have
E=E0ei(kx1-wt)+E0ei(kx2-wt) with x1,x2 the paths from each of the two slits to a point on the screen.
Write each path in terms of a common path x starting midway between the slits, so x1 and x2 both have a path length difference of δ/2 relative to x. Let x1 be the shorter path.
Then x1=x-δ/2 and x2=x+δ/2, and
Factoring out the common terms gives

This is where this issue is. First it says that the intensity is proportional to E*E whilst later it says it is proportional to E*E/2.

This would give either
4E02cos2(kδ/2)->I=4I0cos2(kδ/2), or
Now the first of these is the expression I always see, so must be right.

I’m a little confused by this. I feel like I might be missing something with regards to time averaging the time varying term as in the E*E/2 method. Can somebody clear this up, thanks 🙂

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