Consider the integrator and check for time invariance
Note that there are different kinds of (causal) integrators. The one in your question with a fixed finite lower integration limit is not time-invariant, as you've shown yourself (in the comments).

Other variants are

$$y(t)=\int_{-\infty}^tx(\tau)d\tau\tag{1}$$

and

$$y(t)=\int_{t-T}^tx(\tau)d\tau\tag{2}$$

Both of these integrators are time-invariant, which is straightforward to show.