Your function = X^2
Your Derivative will = 2X
Your Anti Derivative will = 1/3*X^3
An anti derivative is basically, "What function did I have before to get this derivative -- in this case, x^2?"
I don't think there is a huge difference between an integral and anti derivative because if I say, "I'm integrating this function." or "I'm taking the anti derivative of this function." They mean the same thing.
Integrals and anti derivatives share this sign:
If you get...
"The integral from x=1 to x=4 on the graph x^2," then that means you're finding the AREA UNDER THE CURVE X^2 between x=1 and x=4. It's not as simple as flipping a derivative. For this kind of problem, you'll need to take the anti derivative of x^2 and continue on with the Second Fundamental Theorem of Calculus.
Someone else can probably explain it better than me.