The fundamental theorem of calculus describes the relationship between differentiation and integration. The first part of the theorem states that a definite integral of a function can be evaluated by computing the indefinite integral of that function. The second part of the theorem states that differentiation is the inverse of integration, and vice versa.

Theorem

First Statement

Suppose $F$ is a function such that $F'(x)=f(x)$ exists and is continuous on $[a,b]$. Then

Suppose that $f$ is continuous on $[a,b]$ and $\begin{align}F(x)=\int_{\large{a}}^{\large{x}} f(t)\;\mathrm{d}t.\end{align}$ Then $F$ is differentiable on $(a,b)$ and

\[F'(x)=f(x).\]

Corollary

Suppose that $f$ is continuous on $[a,b]$. There there is a function $F$ on $[a,b]$ such that $F$ is differentiable on $(a,b)$ and $F'(x)=f(x)$.