Home  18.013A  Chapter 19 


There is a difficulty with finding antiderivatives akin to the problem of defining the inverse to a function that takes on some of its values at more than one argument each. Knowing that the original function attains one of those values does not determine the inverse function; additional information in the form of an additional condition is necessary to distinguish among the more than one possible original arguments that could be called the inverse function under such circumstances.
In going from the derivative to the function we must contend with the fact that a constant has zero derivative: thus we can add any constant to any possible antiderivative and get another equally valid one.
Which is really to say that to determine an antiderivative completely you must add additional information. In particular the value of the function at any one argument will be enough to determine it from its derivative over its domain.
Thus we must recognize the fact that without an additional condition on f, there is no one single antiderivative f for given g; we can find an antiderivative, or describe all antiderivatives (in which case you should add +c to any one antiderivative) but it is not quite right to use the terminology "the antiderivative of g" until an additional condition has been specified.
(I recall that when I studied calculus in ancient times we had to play a game reminiscent of "Simon Says" with +c; if the question at hand was phrased one way the answer had to have a +c; otherwise not. As I recall from even longer ago, it is very easy to distract a person to lose at Simon Says and it is equally easy to forget to write +c when it is called for.)
