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} $\square$