# The Fundamental Theorem of Calculus

# Antiderivatives

The FTC tells us how to build antiderivatives for any continuous function we like. No user manuel required. There are some things we need to know about these antiderivatives.

\item If we define an antiderivative of *f*(*x*) by

then

because we're integrating over an interval of length 0.

1) If we define two antiderivatives of *f*(*x*) by

where *a* < *b*, then the difference between *F*_{b(x)} and *F*_{a(x)} is

which is a constant.

2) Define an antiderivative of *f*(*x*) by

Since the derivative of a constant is 0, we can add any constant *c* that we like to *F*(*x*) and get a new antiderivative of *f*(*x*). That is, the function

has *G*'(*x*) = *f*(*x*).

Using the information above, we can build antiderivatives with various properties.