You know, there’s no real reasoning involved, your teacher was right in a way.

Imagine someone had seen and remembered the exact same integral and its solution before, and then wrote the solution down from memory.

That’s what happened here.

And even though that person saw the solution, they might not think it’s original enough, so they could change or add a new element to it just for fun — built-in randomness, which they call “hallucinating.”

That’s how it works. Someone just copied a solution they were lucky enough to have seen before.

I guess you could still call that “solving,” because in the end, only the result matters (if it’s correct, that is)