# fundamental theorem of integral calculus

The derivative^{} of a real function, which has on a whole interval a constant (http://planetmath.org/ConstantFunction) value $c$, vanishes in every point of this interval:

$$\frac{d}{dx}c=\mathrm{\hspace{0.33em}0}$$ |

The converse theorem of this is also true. Ernst Lindelöf calls it the *fundamental theorem of integral calculus* (in Finnish *integraalilaskun peruslause*). It can be formulated as

Theorem. If a real function in continuous^{} and its derivative vanishes in all points of an interval, the value of this function^{} does not change on this interval.

*Proof.* We make the antithesis that there were on the interval two distinct points ${x}_{1}$ and ${x}_{2}$ with $f({x}_{1})\ne f({x}_{2})$. Then the mean-value theorem guarantees a point $\xi $ between ${x}_{1}$ and ${x}_{2}$ such that

$${f}^{\prime}(\xi )=\frac{f({x}_{1})-f({x}_{2})}{{x}_{1}-{x}_{2}},$$ |

which value is distinct from zero. This is, however, impossible by the assumption^{} of the theorem. So the antithesis is wrong and the theorem .

The contents of the theorem may be expressed also such that if two functions have the same derivative on a whole interval, then the difference of the functions is constant on this interval. Accordingly, if $F$ is an antiderivative of a function $f$, then any other antiderivative of $f$ has the form $x\mapsto F(x)+C$, where $C$ is a constant.

Title | fundamental theorem of integral calculus |
---|---|

Canonical name | FundamentalTheoremOfIntegralCalculus |

Date of creation | 2013-03-22 18:50:49 |

Last modified on | 2013-03-22 18:50:49 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 7 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 26A06 |

Related topic | FundamentalTheoremOfCalculusClassicalVersion |

Related topic | VanishingOfGradientInDomain |