Fundamental theorem of calculus

The fundamental theorem of calculus forms a link between the fields of differential calculus and integral calculus.

Part one of the theorem

If $f(x)$ is continuous on an interval $I$, the function

$$F(x)={\int }_{x_0}^{x}f(t)dt$$

is an antiderivative of $f(x)$ for any choice of $x_0\in I$. In other words:

$$\frac{d}{dx}{\int }_{x_0}^{x}f(t)dt=f(x)$$

Part two of the theorem

If $F(x)$ is an antiderivative of a function $f(x)$, then:

$${\int }_a^{b}f(x)dx=F(b)-F(a)$$

i.e., the definite integral gives total net change.