I think this system is not time-invariant, but I'm not really sure how to plug in a couple test cases to check. The system is:
$x(t)$ -->(S)--> $y(t) = \int_{-\infty}^{3t}x(\tau) d\tau$
Without actually doing a proof, could I possibly plug in an impulse followed by a shifted impulse to show that it's not time-invariant? What would this look like?
If $y(t)$ is the response to an input signal $x(t)$
$$y(t)=\int_{-\infty}^{3t}x(\tau)d\tau\tag{1}$$
then the system is time-invariant if the response to $x(t-T)$ equals $y(t-T)$. The response to $x(t-T)$ is
$$\int_{-\infty}^{3t}x(\tau-T)d\tau=\int_{-\infty}^{3t-T}x(\tau)d\tau=y(t-T/3)\neq y(t-T)$$
Consequently, the system is not time-invariant, but time varying.