fundamental theorem of calculusFundamentalTheoremOfCalculusClassicalVersion2004-03-01 10:55:532008-04-06 03:33:41Theorem% this is the default PlanetMath preamble. as your knowledge
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% define commands hereLet $f\colon[a,b]\to \mathbf R$ be a continuous function, let $c\in[a,b]$ be given
and consider the integral function $F$ defined on $[a,b]$ as
\[
F(x)= \int_c^x f(t)\, dt.
\]
Then $F$ is an antiderivative of $f$ that is,
$F$ is differentiable in $[a,b]$ and
\[
F'(x)=f(x)\qquad \forall x\in [a,b].
\]
The previous relation rewritten as
\[
\frac{d}{dx} \int_c^x f(t)\, dt = f(x)
\]
shows that the differentiation operator $\frac{d}{dx}$ is the inverse of the integration operator $\int_c^x$. This formula is sometimes called Newton-Leibniz formula.
On the other hand if $f\colon[a,b]\to \mathbf R$ is a continuous function
and $G\colon[a,b]\to \mathbf R$ is any antiderivative of $f$, i.e.\ $G'(x)=f(x)$ for all $x\in[a,b]$, then
\begin{equation}\label{eq:barrow}
\int_a^b f(t) \, dt = G(b)-G(a).
\end{equation}
This shows that up to a constant, the integration operator is the inverse of the derivative operator:
\[
\int_a^x D G = G - G(a).
\]
\section*{Notes}
Equation~\eqref{eq:barrow} is sometimes called ``Barrow's rule'' or ``Barrow's formula''.