Definite Integral of Odd Function

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Theorem

Let $f$ be an odd function with a primitive on the closed interval $\left[{-a \,.\,.\, a}\right]$, where $a > 0$.


Then:

$\displaystyle \int_{-a}^a f \left({x}\right) \ \mathrm d x = 0$


Proof

Let $F$ be a primitive for $f$ on the interval $\left[{-a \,.\,.\, a}\right]$.

Then, by Sum of Integrals on Adjacent Intervals for Integrable Functions, we have:

\(\displaystyle \int_{-a}^a f \left({x}\right) \ \mathrm d x\) \(=\) \(\displaystyle \int_{-a}^0 f \left({x}\right) \ \mathrm d x + \int_0^a f \left({x}\right) \ \mathrm d x\)

Therefore, it suffices to prove that:

$\displaystyle \int_{-a}^0 f \left({x}\right) \ \mathrm d x = - \int_0^a f \left({x}\right) \ \mathrm d x$


To this end, let $\phi: \R \to \R$ be defined by $x \mapsto -x$.

From Derivative of Identity Function and Derivative of Constant Multiple, for all $x \in \R$, we have $\phi' \left({x}\right) = -1$.

Then, by means of Integration by Substitution, we compute:

\(\displaystyle \int_{\phi \left({a}\right) }^{\phi \left({0}\right)} f \left({x}\right) \ \mathrm d x\) \(=\) \(\displaystyle \int_a^0 f \left({-u}\right) \left({-1}\right) \ \mathrm d u\)
\(\displaystyle \) \(=\) \(\displaystyle \int_0^a f \left({-u}\right) \ \mathrm d u\) Definition of integral
\(\displaystyle \) \(=\) \(\displaystyle \int_0^a f \left({-x}\right) \ \mathrm d x\) Renaming integration variable
\(\displaystyle \) \(=\) \(\displaystyle -\int_0^a f \left({x}\right) \ \mathrm d x\) $f$ is an odd function

This concludes the proof.

$\blacksquare$


Also see


Sources