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# 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”.

## Mathematics Subject Classification

26A42*no label found*

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## Comments

## Why does this entry exist?

We already have an entry on the Fundamental Theorem of Calculus which covers both fundamental theorems of calculus. Its generality is slightly less in some ways (requires absolute continuity) but this could be fixed. What is the need for a second entry?

## Re: Why does this entry exist?

Actually the old entry is more general! In fact it is the integral function F which is supposed to be only absolutely continuous not the integrand f. This is not the classical result and, to my knowledge, can only be proven using the Lebesgue Differentiation Theorem. See the discussion on "Errata and Addenda".

## Re: Why does this entry exist?

Right. So why this entry? If it's just that the proof is easier, it makes more sense to add a special-case proof attached to the other entry.

Perhaps this is intended as a simplified version for people who would be confused by the real result? If so, it should include the converse (especially since it asserts that the derivative and integral operators are inverses but only shows they are one-sided inverses). The converse is easy for nice enough (C^1 will do) functions.

## Re: Why does this entry exist?

Yes, this theorem is much more easier to prove. Also this is the statement which is labeled "Fundamental Theorem of Calculus" in every Calculus book.

To prove the general theorem you have to introduce the Lebesgue measure, L^p spaces, Sobolev spaces and so on... Also the general theorem holds for Lebesgue integrals but not for Riemann integrals.

## Re: Why does this entry exist?

Makes sense.

Perhaps this explanation belongs in the entry?