Fundamental Theorem of Calculus
As the name suggests, the Fundamental Theorem of Calculus (FTC) is an important theorem. The theorem connects integrals and derivatives. There are two versions.
Version 1
Define
where f(x) is a continuous function. (This assumption can be weakened.) In other words, F(t) is simply the area under the f(x) curve from a to t. The Fundamental Theorem of Calculus states
There is an analogous result for indefinite integrals. Let
Then
Version 2
The second version of the Fundamental Theorem of Calculus states that
This last formula can also be expressed in terms of an indefinite integral:
where C is a constant.