.
Summary
We've been taking derivatives of functions: from a function, we create a new function, called the derivative. Now we want to undo that, just like we use inverse functions to undo one another (think of logarithms and exponential functions). If you're given a function f, can you interpret it as a derivative? That is, can you find a function F that has that function f as its derivative? So here's the deal:
Before: given f(x), find .
Now: given 
, find F(x).
Only we're going to be given f(x); think of it as a derivative, f(x)=F'(x); and try to find F.
Now it turns out that this is harder than it seems: the process of finding derivatives is very mechanical, and it turns out that the inverse process is not so mechanical. Let's see how far we can get....
Antiderivative: If F'(x)=f(x), then F(x) is an antiderivative of f(x).
Note that we say ``an'' antiderivative: it turns out that it's not unique. Note also that we're using small letters for functions we're thinking of as derivatives, and capitals for those we're thinking of as antiderivatives.
If F(x) and G(x) are both antiderivatives of a function f(x) on an interval, then
for some constant C.
So if you can find an antiderivative F, then you can write them all as F(x)+C, where C is an arbitrary constant. This represents a family of functions. We have a special (and quite unusual) notation for this family:
is called the indefinite integral of f; the symbol 
is called
the integral sign, and f(x) is called the integrand of the
integral. So
for any real number C. The x inside the integral is called a dummy variable: so
works just as well!
Some examples:
As in the case of derivatives, we can write up a bunch of general rules for integrals:
Some examples: