Theorem 2.1. The Fundamental Theorem of Calculus.
If \(f(x)\) is continuous, and the derivative of \(F(x)\) is \(f(x)\text{,}\) then
\begin{equation*}
\definiteintegral{a}{b}{f(x)}{x}=F(b)-F(a)
\end{equation*}
Section 2 The Fundamental Theorem
There is a remarkable theorem:
1
And fortunately we do not need to try to write it in the margin!
Proof.
Left to the reader.
You will find almost nothing about all this in the article [2], nor in the book [1], since they belong in some other article, but we can cite them out-of-order for practice anyway.
When we are writing we do not always know what we want to cite, or just where subsequent material will end up. For example, we might want a citation to
[provisional cross-reference: some textbook about the FTC] or we might want to reference a later [provisional cross-reference: chapter about DiffEq's, and an_underscore].We can also embed “todo” s in the source by making an XML comment that begins with the four characters
todo, and selectively display them, so you may not see the one here in the output you are looking at now. Or maybe you do see it?Because a definite integral can be computed using an antiderivative, we have the following definition.
Definition 2.2.
Suppose that \(\frac{d}{dx}F(x)=f(x)\text{.}\) Then the indefinite integral of \(f(x)\) is \(F(x)\) and is written as
\begin{equation*}
\int\,f(x)\,dx=F(x)\text{.}
\end{equation*}
