Working out integration of basic signal functions As the title suggests , I'm having problems working out the integration of some basic signal functions. Shown below are the questions and my workings. For some reason I'm just unable to get the answer.

For a) and c) , I know that for u(t) is basically 0 before t=0 , thus I changed the limits for the integration. I did the same thing for u(t-1)

Thanks for the help!
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.