Part 1

Let $a

$$ F(x) = \int_{a}^x f(t) dt $$

Note: the variable inside the integral is $t$ (dummy variable), while the variable defining the function is $x$ (the upper limit) Then $F(x)$ is continuous on $[a,b]$, differentiable on $(a,b)$ and its derivative is:

$$ F'(x) = f(x) $$

Part 2

Let $a

Then:

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

where $F(x)$ is any antiderivative of $f(x)$.