fundamental theorem of integral calculus

The derivativePlanetmathPlanetmath of a real function, which has on a whole interval a constant ( value c, vanishes in every point of this interval:

ddxc= 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 continuousMathworldPlanetmathPlanetmath and its derivative vanishes in all points of an interval, the value of this functionMathworldPlanetmath does not change on this interval.

Proof.  We make the antithesis that there were on the interval two distinct points x1 and x2 with  f(x1)f(x2).  Then the mean-value theorem guarantees a point ξ between x1 and x2 such that


which value is distinct from zero.  This is, however, impossible by the assumptionPlanetmathPlanetmath 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 xF(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