Theorem. Let f be a function which is continuous on the interval [a, b].
  1. Let F be an indefinite integral or antiderivative of f. Then

  2. The function

    is an indefinite integral or antiderivative of f. That is,

    A'(x) = f(x)

     

The first part of this theorem tells us how to evaluate a definite integral provided that f has an indefinite integral.

The second part of the theorem tells us that f has an indefinite integral. The only remaining problem is actually finding a formula for the indefinite integral which we can easily evaluate.



In general, we will not be able to find a "formula" for the indefinite integral of a function. However, using the second part of the Fundamental Theorem, we are still able to draw the graph of the indefinite integral:


Check out the proof of the Fundamental Theorem:


The fundamental theorem of calculus and the chain rule: