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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.

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    $\begingroup$ You have the Laplace transform of $g(t)*f(t)$ where $g(t)=e^t\sin t$. So the Laplace transform of this is $\mathcal{L}[g(t)]\mathcal{L}[f(t)]$ where $\mathcal{L}[\cdot]$ denotes a Laplace transform. You are left to compute $\mathcal{L}[e^t\sin t]$. $\endgroup$ – John Doe Dec 11 '17 at 3:03
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For functions $f(x)$ and $g(x)=e^x\sin x$ then $${\cal L}\int^t_0 e^{t-\tau} sin(t-\tau) f(\tau) d(\tau)={\cal L}(f*g)={\cal L}(f)\dfrac{1}{(s-1)^2+1}$$

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