Fundamental Theorem of Calculus

There are two parts to it

Suppose that the function $f$ is continuous on an interval $I$ containing the point $a$ .

Let the function $F$ be defined on $I$ :

$F(x) = \int_{a}^{x} F'(t) dt = \int_{a}^{x} f(t) dt$

Then F is differentiable on $I$ , and $F'(x) = f(x)$ . Thus $F$ is an antiderivative of $f$ on the interval $I$ .

If $G(x)$ is any antiderivative of $f(x)$ on $I$ such that $G'(x) = f(x)$ on $I$ , then for any $b$ in $I$ we have:

$\int_{a}^{b} f(x) dx = G(b) - G(a)$