I was missing the principle that when you want to say "in the month of XXXX" you use "au mois de XXXX". It follows then to use Aux when multiple months involved. I found that on this page: http://www.frenchlearner.com/vocabulary/months-of-the-year/
Here is the quote from the page: If you want to say “In + a month” in French the grammatical construction is “Au mois de _.” in the blank you fill in the month. The literal translation for this radical structure is “In the month of __”. for example, if you want to say “I was born in January” you would say, “Je suis né(e)au mois de janvier.
I could be wrong but it seems like "le mois d'avril" does not require another article in the same say that "une tasse de thé" does not require another article. Try this link for explanations using "de": http://french.about.com/od/grammar/a/preposition_de.htm
You can't be "in" a month like "in" a house, but you still use "in" for months in English, don't you? And I think you don't always replace it with "during". I was born in September, not during September, for example. Here without context you can't say if "during" is a reasonable replacement of "in", but anyway, the fact is that there is a separate word for "during" in French, and it is not used in this sentence.
I had "In the month of April and of May" rejected (I was told it should be "months") which I think is wrong. It's on the borderline between "In the months of April and May" (definitely plural) and "In the month of April and in the month of May" (definitely singular). I think if you repeat the "of" singular should be okay (basically read like inline dot-points).
Maybe it's just my rotten English.
No, sorry. "In the month of April and of May" is incorrect.
I'm tired, and can't think of the correct grammatical construction, but in simple terms, in this example, you can't split a single noun between two separate nouns.
Take this as another example:
David and Sue each have a car.
In the car of David and of Sue. (Incorrect because they aren't sharing the one car).
In the car of David and Sue. (same reason as above).
In the cars of David and Sue. (Tells us there is more than one car)
In the car of David and in the car of Sue. (Points out that we are talking about two cars, each owned by different people).
So......... if you are talking about two things, you treat them as a plural ("In the months etc.") or if you wish to talk about them individually, you describe them as two separate things (In the month of April and in the month of May).
Because that would shift the meaning. "During" implies something that takes place over all or most of that two-month period - "We always take our vacation during the months of April and May." "In" is much less about duration - "All four of my sisters' birthdays are in the months of April and May."