The formal definition of an indefinite integral giving the antiderivative is if , then we would have that . In words this just basically means that the antiderivative of a function is ANY function that when you take the derivative of it you get the orginal function. That is the need for the +C. For consider . We do not know what C is but no matter what it is when I take the derivative of I get . To illustrate that if you asserted that and only I would state that is one of them since . But I would then say what about ? Because we have that

The difference between a definite integral an indefinite integral is very simple. An indefinite as was shown above is a function, more accurately a set of functions, but a function nonetheless. But a definite integral is just a number.

To see this we need to accept the First Fundamental Theorem of Calculus which states that . You may interpret this as evaluating the antiderivative of the function at the upper bound and substracting the antiderivative at the lower bound. But you might ask what about the C? Well the gets nicely taken care of. Lets not just state the theorem but lets actually play it out a little more. Assume for a second that . Well we know that . So now we rewrite our definite integral as follows

And since we know that when we evaluate a function at a number we just get a number.

Does that help at all?