# fundamental theorem of calculus

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

$$F(x)={\int}_{c}^{x}f(t)\mathit{d}t.$$ |

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

$${F}^{\prime}(x)=f(x)\mathit{\hspace{1em}\hspace{1em}}\forall x\in [a,b].$$ |

The previous relation^{} rewritten as

$$\frac{d}{dx}{\int}_{c}^{x}f(t)\mathit{d}t=f(x)$$ |

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

$${\int}_{a}^{b}f(t)\mathit{d}t=G(b)-G(a).$$ | (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 |