Determining if a system involving integrals is time varying or time-invariant
The fixed bound at 0 in $\int_0^t$ indeed suggests a time-variance.You only need a counter-example for $S_1$, like the constant $x(t) = 1$.  $y(t+1) = t+1$, while $S_1(x(t+1)) = S_1(x(t)) = y(t)$.

For $S_2$, the two bounds moving at the same time are promising, and a standard variable change shows the time invariance.