Can I use convolution to find the Laplace transform

Question

The question states:

Solve the Laplace transform of the following integral $$ \int^t_0 e^{t-\tau} sin(t-\tau) f(\tau) d(\tau) $$

It screams convolution to me, but I can't seem to figure out how to use it. Is it possible to use it in this case, or do I just use other methods.

Answer 1

For functions $f(x)$ and $g(x)=e^x\sin x$ then $${\cal L}\int^t_0 e^{t-\tau} sin(t-\tau) f(\tau) d(\tau)={\cal L}(f*g)={\cal L}(f)\dfrac{1}{(s-1)^2+1}$$