Change $p \rightarrow -p$ $ d^{3}p$ invariant

Question

I am unsure how to change of variables $p \rightarrow -p$ is valid. I would have thought the integral would pick up a - on the $d^{3}p$?

David Tong Notes https://i.stack.imgur.com/zerx0.png

Answer 1

The integral picks up two minus signs - one from the substitution, and another from changing the direction of integration. For a simpler example: $$\int_{-\infty}^\infty f(x) \, dx = \int_{\infty}^{-\infty}-f(-t)\,dt = \int_{-\infty}^\infty f(-t) \, dt.$$