# fundamental theorem of calculus

Let $f\colon[a,b]\to\mathbf{R}$ be a continuous function, let $c\in[a,b]$ be given and consider the integral function $F$ defined on $[a,b]$ as

 $F(x)=\int_{c}^{x}f(t)\,dt.$

Then $F$ is an antiderivative of $f$ that is, $F$ is differentiable in $[a,b]$ and

 $F^{\prime}(x)=f(x)\qquad\forall x\in[a,b].$

The previous relation rewritten as

 $\frac{d}{dx}\int_{c}^{x}f(t)\,dt=f(x)$

shows that the differentiation operator $\frac{d}{dx}$ is the inverse of the integration operator $\int_{c}^{x}$. This formula is sometimes called Newton-Leibniz formula.

On the other hand if $f\colon[a,b]\to\mathbf{R}$ is a continuous function and $G\colon[a,b]\to\mathbf{R}$ is any antiderivative of $f$, i.e. $G^{\prime}(x)=f(x)$ for all $x\in[a,b]$, then

 $\int_{a}^{b}f(t)\,dt=G(b)-G(a).$ (1)

This shows that up to a constant, the integration operator is the inverse of the derivative operator:

 $\int_{a}^{x}DG=G-G(a).$

## Notes

Equation (1) is sometimes called “Barrow’s rule” or “Barrow’s formula”.

 Title fundamental theorem of calculus Canonical name FundamentalTheoremOfCalculus Date of creation 2013-03-22 14:13:27 Last modified on 2013-03-22 14:13:27 Owner paolini (1187) Last modified by paolini (1187) Numerical id 13 Author paolini (1187) Entry type Theorem Classification msc 26A42 Synonym Newton-Leibniz Synonym Barrow’s rule Synonym Barrow’s formula Related topic FundamentalTheoremOfCalculus Related topic FundamentalTheoremOfCalculusForKurzweilHenstockIntegral Related topic FundamentalTheoremOfCalculusForRiemannIntegration Related topic LaplaceTransformOfFracftt Related topic LimitsOfNaturalLogarithm Related topic FundamentalTheoremOfIntegralCalculus