Let $[- a, a] \subseteq \mathbb{R}$ be a D544: Closed real interval such that

(i)	$f : [- a, a] \to \mathbb{R}$ is a D1760: Riemann integrable real function on $[- a, a]$
(ii)	$f$ is an D5480: Even real function

Applying results

(i)	R4288:
(ii)	R3061: Real Riemann integral over a reflected interval

we have \begin{equation} \begin{split} \int^a_{-a} f(x) \, d x & = \int^a_0 f(x) \, d x + \int^0_{-a} f(x) \, d x \\ & = \int^a_0 f(x) \, d x + \int^a_0 f(-x) \, d x \\ & = \int^a_0 f(x) \, d x + \int^a_0 f(x) \, d x \\ & = 2 \int^a_0 f(x) \, d x \end{split} \end{equation} $\square$