Even and odd functions

The symmetries of a function simplify the computation of an integral over a symmetric interval $[-a,a]$.

• If $f$ is even on $[-a,a]$: $\displaystyle\int_{-a}^a f\,dx = 2\int_0^a f\,dx$.
• If $f$ is odd on $[-a,a]$: $\displaystyle\int_{-a}^a f\,dx = 0$.

It is also useful to remember how parities combine in a product:

$\times$	even	odd
even	even	odd
odd	odd	even

Recognising the parity of the integrand before computing can make the integral vanish (odd case) or halve the work (even case).

Links

Topics: Integrale
Concepts: Funzione dispari · Funzione pari · Integrale definito
Skills: Integrare · Usare formule