## Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is one of the most amazing and important theorems in analysis. It is a non-trivial result that links the concept of area and gradient, two seemingly unrelated concepts.

## Fundamental Theorem of Calculus

The first part deals with the derivative of an antiderivative, while the second part deals with the relationship between antiderivatives and definite integrals.

## First part

Let $f$ be a continuous real-valued function defined on a closed interval $[a,b]$. Let $F$ be the function defined, for all $x$ in $[a,b]$, by $\displaystyle F(x)=\int_a^x f(t)\,dt.$

Then $F$ is uniformly continuous on $[a,b]$, differentiable on the open interval $(a,b)$, and $\displaystyle F'(x)=f(x)$ for all $x$ in $(a,b)$.

## Second part

Let $f$ and $F$ be real-valued functions defined on $[a,b]$ such that $F$ is continuous and for all $x\in (a,b)$, $\displaystyle F'(x)=f(x).$

If $f$ is Riemann integrable on $[a,b]$, then $\displaystyle \int_a^b f(x)\,dx=F(b)-F(a).$