fundamental theorems of calculus for Lebesgue integrationFundamentalTheoremOfCalculus2002-02-24 19:50:492007-06-24 01:19:02Theorem% this is the default PlanetMath preamble. as your knowledge
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% define commands hereLoosely, the \emph{Fundamental Theorems of Calculus} serve to demonstrate that integration and differentiation are inverse processes. Suppose that $F(x)$ is an absolutely continuous function on an interval $[a,b]\subset\mathbb{R}$. The two following forms of the theorem are equivalent.
{\bf First form of the Fundamental Theorem:}
There exists a function $f(t)$ Lebesgue-integrable on $[a,b]$ such that for any $x\in [a,b]$, we have $F(x)-F(a)=\int_a^x f(t) dt$.
{\bf Second form of the Fundamental Theorem:}
$F(x)$ is differentiable almost everywhere on $[a,b]$ and its derivative, denoted $F'(x)$, is Lebesgue-integrable on that interval. In addition, we have the relation $F(x)-F(a)=\int_a^x F'(t)dt$ for any $x\in [a,b]$.