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# fundamental theorems of calculus for Lebesgue integration

Loosely, the *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.

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 $F(x)-F(a)=\int_{a}^{x}f(t)dt$.

Second form of the Fundamental Theorem:

$F(x)$ is differentiable almost everywhere on $[a,b]$ and its derivative, denoted $F^{{\prime}}(x)$, is Lebesgue-integrable on that interval. In addition, we have the relation $F(x)-F(a)=\int_{a}^{x}F^{{\prime}}(t)dt$ for any $x\in[a,b]$.

## Mathematics Subject Classification

26-00*no label found*

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