Assuming $\mathscr{L}(x(t)) = X(s)$ then I would like to know $Y(s)$, where $y(t)$ is
$y(t) = \int_{t-\Delta}^{t} x(\tau) d\tau $
I guess one can start in the following way (using the two-side laplace transform):
$y(t) = \int_{t-\Delta}^{t} x(\tau) d\tau = \int_{-\infty}^{t} x(\tau) d\tau - \int_{-\infty}^{t-\Delta} x(\tau) d\tau $
Hence
$Y(s) = X(s)/s - Y_{unknown} (s) $
So what is the $Y_{unknown}(s)$?
Thank you in advance