Integral of an odd function over a symmetric domain vanishes

I show that the antiderivative of an odd function is an even function by using the reflection substitution, and thus the integral of an odd function over a symmetric domain vanishes. This simple theorem comes in handy when breaking a complicated integral into simpler ones, some of which can be shown to vanish.

[muse]

integral dx x? what is that? integral x dx is the right expression. More. integral dx=c, integral dx x=cx