# integrals of even and odd functions

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

• even function, then  $\displaystyle\int_{-a}^{a}{f(x)}\,dx\;=\;2\int_{0}^{a}f(x)\,dx$,

• odd function, then  $\displaystyle\int_{-a}^{a}f(x)\,dx\;=\;0.$

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

 $I\;:=\;\int_{-a}^{a}f(x)\,dx\;=\;\int_{-a}^{0}f(t)\,dt+\int_{0}^{a}f(x)\,dx.$

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