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Fundamental theorem of calculus

Fundamental theorem of calculus

Part 1 Let f(x) be a continuous real valued function on the interval $[a,b]$, we define $F(x)$ by $$ F(x) = \int_a^x f(t)dt $$ It follows that $F'(x) = f(x)$ for all $x\in (a,b)$. (We call $F(x)$ an antiderivative of $f(x)$)

Part 2 Let $f(x)$ be a real valued function on $[a,b]$ and let $F(x)$ be an antiderivative of f(x) then $$ \int_a^b f(x)dx = F(b) - F(a) $$

Note that in the second part, we do not require that $f(x)$ is continuous, only that it is Riemannian Integrable, so we can have piecewise continuous functions.