Can someone explain the difference between an antiderivative of f(x) an the indefinite integral of f(x)?

They are pretty much the same thing. It really just depends on how you use them. You usually say that your are taking an antiderivative when you are given the derivative and are finding the original function. You usually say integral when you have a function and you are just doing something with it that involves integrating it (ie find the area under the curve, etc.).