### Fundamental Theorem of Calculus

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

$\int_{\large{a}}^{\large{b}} f(x)\;\mathrm{d}x=F(b)-F(a).$

###### Second Statement

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

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