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If you want to get a flavor of the article without being too much of a math person, you can watch the 3Blue1Brown video linked in the article, which captures a similar surprise that for certain values a formula maps exactly to a "round" quantity, then starts to diverge after a while.

https://www.youtube.com/watch?v=851U557j6HE

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My ignorant intuition is since it's the area between sin(x)/x and the x axis, but as you get further into infinity the 1/x still keeps getting smaller slowly but sin(x) always is the same magnitude, the integral gets slightly further away the closer you get to infinity and that slight difference adds up?

Yes but the question is why the value is exact for small N.

that borwein integral formula is one of the first things baez mentions

The comment you replied to mentions that the video is linked in the article.

so it does, i'm an idiot, thanks

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