Is it correct to say I've got it?
"I've got it" (I have got it) is ok to mean one thing.

"I got it" is ok to mean something else.

It depends on what you are talking about.
They are different forms.

Maybe one of the trained English grammar teachers here can elaborate?
I'm sorry, I don't understand your answer.
What two different things do they mean?
Students: We have free audio pronunciation exercises.
Do you have the new 'Black Star Liner' CD?

Yes, I've got it. (present)

Oh, when did you get it?

I got it at the shop last week. (past)
Why can't we just say ' I have it' ?
Yes, it is correct to say "I've got it". This statement can mean "I have it" or "I have solved a problem", and maybe more.

One could easily fill a page writing about the use of "got" (and the differences between British and American usage).
Students: Are you brave enough to let our tutors analyse your pronunciation?
instead of saying "I have got... ", we should say "I have gotten..." . Is that correct?
well i think i have got it can be bit more of like a surity that i have surely got it.On the other hand i have it is bit casual
Is it correct to say "I got a pen" which means "I have a pen" and "I got to go" which means "I have to go"?
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