How does one prove the following integral identity, where $P_n(x)$ is the $n$th Legendre polynomial? $$ \int_0^1 P_n(2t^2-1) dt = \frac{(-1)^n}{2n+1} $$

### Notes & Background

A variant of this appears in, for instance, Erdelyi et al "Higher transcendental functions" 10.10(49), but with nothing in the way of explanation.

This comes up in harmonic analysis on $U(3)$, when comparing Gelfand-Tseltin bases associated to different choices of nested sequences $U(3) \supset U(2) \supset U(1)$.

Eventually, I'll be looking for a $q$-analogue, related to harmonic analysis on $U_q(3)$, so a proof that will transport well would be my true desire.