integrals of even and odd functions

Theorem.  Let the real function $f$ be Riemann-integrable (http://planetmath.org/RiemannIntegrable) on  $[-a,a]$.  If $f$ is an

• even function, then  ${\displaystyle {\int }_{-a}^a}f(x)𝑑x=\mathrm{\hspace{0.33em}2}{\displaystyle {\int }_0^a}f(x)𝑑x$,

• odd function, then  ${\displaystyle {\int }_{-a}^a}f(x)𝑑x=\mathrm{\hspace{0.33em}0}.$

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

$$I:={\int }_{-a}^{a}f(x)𝑑x={\int }_{-a}^{0}f(t)𝑑t+{\int }_0^{a}f(x)𝑑x.$$

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

$$I={\int }_a^{0}f(-x)(-dx)+{\int }_0^{a}f(x)𝑑x={\int }_0^{a}f(-x)𝑑x+{\int }_0^{a}f(x)𝑑x.$$

Using then the definitions of even (http://planetmath.org/EvenoddFunction) ($+$) and odd (http://planetmath.org/EvenoddFunction) ($-$) function yields

$$I={\int }_0^a(\pm f(x))𝑑x+{\int }_0^{a}f(x)𝑑x=\pm {\int }_0^{a}f(x)𝑑x+{\int }_0^{a}f(x)𝑑x,$$

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