Convolution integral $x_1(t)\ast x_2(-t)$ sign change

Question

Sincerely ask a very fundamental problem. We know $$x_1(t)\ast x_2(t) = \int_{-\infty}^{\infty}x_1(\tau)x_2(t-\tau)d\tau.$$

My question is what is $x_1(t)\ast x_2(-t)$?

Should it be $$x_1(t)\ast x_2(-t) = \int_{-\infty}^{\infty}x_1(\tau)x_2(-t-\tau)d\tau$$

or

$$x_1(t)\ast x_2(-t) = \int_{-\infty}^{\infty}x_1(\tau)x_2(-t+\tau)d\tau$$

How to strictly argue the correct one?

Thanks!

Answer 1

I think this should be pretty straightforward if you consider $y(t) = x_2(-t)$.