Instructions: Evaluate the given Laplace transform. Do not evaluate the integral before transforming.
Problem Given: $\mathscr{L}\{\int_0^t e^{-\tau} cos\tau d\tau \}$
My Problem: To treat this as a convolution, I think I need to rewrite one of the two functions of $\tau$ as a function of $t-\tau$. My intuition is to use the following trig identity, but this could make things nasty come test time (i.e. tomorrow):
$cos{t-\tau} = {costcos\tau - sintsin\tau}$
$cos\tau = \frac{cos{(t-\tau)} - sintsin\tau}{cost} $
Do I just need to tough it out going this route, or am I approaching this poorly?