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 $F'(x)$ , is Lebesgue-integrable on that interval. In addition, we have the relation $F(x)-F(a)=\int_a^x F'(t)dt$ for any $x\in [a,b]$ .