Laplace Transform Integral

Question

$$\mathscr{L}\left ( \int_0^t e^{t-\tau}\cos(t-\tau)e^{-\tau}d\tau \right )$$

Hi all I am trying to solve this Laplace transform but I dont know if it's correct, please tell me.

Here is my attempt:

Answer 1

You made some mistakes because in the final answer you should only have $s$ variable and not both $t,s$ variables. There are no trig functions in the final answer.

$$W(s)=\mathscr{L}\left ( \int_0^t e^{t-\tau}\cos(t-\tau)e^{-\tau}d\tau \right )$$ You can also use the Theorem of Convolution: $$\mathscr{L}\left ( \int_0^t f(t-\tau)g{(\tau)}d\tau \right )=F(s)G(s)$$ $$W(s)=\mathscr{L}\left ( e^t\cos(t)*e^{-t}\right)$$ Therefore: $$W(s)=\dfrac {s-1}{(s-1)^2+1}\dfrac 1 {s+1}$$ Finally: $$W(s)=\dfrac {s-1}{s^3-s^2+2}$$