integrals of even and odd functions

Theorem.  Let the real function f be Riemann-integrable ( on  [-a,a].  If f is an

  • even functionMathworldPlanetmath, then  -aaf(x)𝑑x= 20af(x)𝑑x,

  • odd function, then  -aaf(x)𝑑x= 0.

Of course, both cases concern the zero map which is both .

Proof. Since the definite integral is additive with respect to the interval of integration, one has


Making in the first addend the substitution  t=-x,dt=-dx  and swapping the limits of integration one gets


Using then the definitions of even ( (+) and odd ( (-) functionMathworldPlanetmath yields


which settles the equations of the theorem.

Title integrals of even and odd functions
Canonical name IntegralsOfEvenAndOddFunctions
Date of creation 2014-03-13 16:17:44
Last modified on 2014-03-13 16:17:44
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Theorem
Classification msc 26A06
Synonym integral of odd function
Synonym integral of even function
Related topic DefiniteIntegral
Related topic ChangeOfVariableInDefiniteIntegral
Related topic ExampleOfUsingResidueTheorem
Related topic FourierSineAndCosineSeries
Related topic IntegralOverAPeriodInterval