Construct the convolution of g(t) and k(t)

I first write

Can I make the same move I would make with the Heaviside step function and rewrite the integral as..

due to the fact that the function would be off while t < 2 ?

If not, can i use the sifting property and just evaluate with t = 2, = . Or do the limits play a part here?

Would I also have to throw the heaviside in after that solution? Resulting in

I know I'm pulling from properties about the Heaviside and Dirac, just a little confused.

Thanks