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

$\displaystyle g(t) = e^t, k(t) = \delta(t-2) $

I first write

$\displaystyle \int_0^t \delta(\tau-2)e^{t-\tau} d\tau $

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

$\displaystyle \int_2^t e^{t-\tau} d\tau $

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

If not, can i use the sifting property and just evaluate $\displaystyle e^{t-\tau}$ with t = 2, = $\displaystyle e^{t-2} $. Or do the limits play a part here?

Would I also have to throw the heaviside in after that solution? Resulting in $\displaystyle e^{t-2}H(t-2) $

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

Thanks