
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: