Sometimes, it's difficult to decide the preposition which fits in, without going with the instincts or "oh yeah, this sounds right"
In the sentence: I've been busy for the past few days, can 'in' substitute 'for'? Which one is correct and why?
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Both "for" and "in" occur with "the past few days".
"for the past few days" suggests something that has continued over that time.
"in the past few days" suggests something or some things that have happened several times in that time period.
Google ngram viewer shows that both expressions are pretty common:
ngram viewer "in/for the last few days"
If you go to that link, you will see a graph of both expressions over time, and at the bottom of the page, there are links to examples of each at Google books.