For fun I’m enrolled in an online computational finance certificate at UW. In one of my homework problems I wanted to use the following fact about the integral of single variable, even functions:

If it’s been a few years since you’ve taken calculus that may not make much sense, but trust me when I tell you that it’s analytically obvious, especially when looking at functions graphically, as this terrible hand drawn image shows:

Intuitively, we know the two red areas are the same, so it seems we should be able to interchange the limits as I described above. Indeed, playing around in Mathematica suggests that this is true. However, I could not find a proof or theorem for this online so perhaps it is rarely used. I decided to prove it myself:

Original equation:

Use u-substitution with and :

Bring minus sign outside integral:

Use the fact that :

By assumption is even so :

Rewrite improper integral:

By Fundamental Theorem of Calculus:

By Fundamental Theorem of Calculus:

Which is exactly the result we were trying to obtain.