Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) demonstrates that differentiation and integration are inverse processes of one another.

title: The Fundamental Theorem of Calculus (Part I)
If $f(x)$ is continous on $[a,b]$, then the function $g(x)$ defined by 
$$g(x) =  \int_{a}^{x}f(t)dt, \qquad \text{for } x \in [a,b]$$
is differentiable on (a,b), and its derivative is $g'(x) = f(x)$
 

If $F(x)$ is an antiderivative of $f(x)$, then every antiderivative of $f(x)$ can be expressed as $F(x)+C$, for some constant $C$.

title: The Fundamental Theorem of Calculus (Part II)
If $F(x)$ is any antiderivative of $f(x)$, then
$$ \int_{a}^{b}f(t)dt = F(b) - F(a)$$