Working out integration of basic signal functions
So you've already figured out the $u(t-T)$ trick:  the unit step (delayed) just changes the limits of integration.

The other trick is that $\delta(t-T)$ just "samples" the value of the rest of the integral's argument. So, for example, $\int x(t) \delta(t-T) dt = x(T)$ where $T$ is somewhere within the integral's limits.