In my textbook, there are two fundamental theorems of Calculus. The first states:
Let $ f $ be integrable on $[a,b]$. For $x\in[a,b]$ put $F(x)=\int^x_a f(t)dt$. Then $ F$ is continuous on $[a,b]$ and furthermore if $ f $ is continuous at a point $x_0$ of $[a,b]$ then $F$ is differentiable at $x_0$ and $F'(x_0)=f(x_0)$.
And the second states:
If $ f $ is integrable on $[a,b]$ and if there is a differentiable function $ F $ on $[a,b]$ such that $F'=f$, then $\int^b_a f(x)dx=F(b)-F(a)$.
My question is; what is the difference between these two? when is each useful?