As part of a proof for Moment Generating Functions being symmetric about 0 for even probability density functions, I see the following line in my textbook:
$$ \int_{-\infty}^0 e^{\epsilon x} f_X(x) dx + \int_0^{\infty} e^{\epsilon x} f_X(x) dx = \int_0^{\infty} e^{-\epsilon x} f_X(-x) dx + \int_{-\infty}^0 e^{-\epsilon x} f_X(-x) dx$$
I am pretty familiar with integrals changing signs when flipping boundaries, but have not seen the above before. What is the justification for that? Thanks.
It's just elementary (Calculus 1) change of variables, e.g. for the first integral let $u=-x$, so (informally) $du=-dx$ and the limits of integration now go from $\infty$ to $0$, hence:
$$ \int_{-\infty}^{0} \exp(\varepsilon x) f_X(x) dx = \int_{\infty}^0 -\exp(-\varepsilon u) f_X(-u)du = \int_{0}^{\infty} \exp(-\varepsilon u) f_X(-u) du $$
Note that the 1st equality here is just the change of variables, and the second one is the "changing signs when flipping boundaries" which you mention.