I could pretend to be available that day, maybe, but it’s mutually beneficial for me to just take a day off.
The next day I’m usually just fine, and I don’t always give more explanation than “taking a sick day”.
SOmetimes it can knock me out for a day, but sometimes after half an hour or so I'm good to go (especially if there is no pain accompanying it).
I've never had to give more explanation than "Sorry, migraine, I need to take rest of day off, see you tomorrow".
One of my worst examples was that I made it to work on the bus, and immediately had to take off. I was pretty nauseous by the time I made it home, and should I have been found throwing up in a bush, nobody would have believed “I have a migraine”. To the drunk tank with me, at 930am.
And that's entirely justified; your health is a private matter. If your employer is suspicious, there's ways and means that don't involve sharing your medical history with either your colleagues or employer.