Find the Laplace of the given function using the definition
$f(t)=e^{-t} \operatorname{sen}(t)$
I know what the answer is according to a sheet that I have of common transforms but I am not 100% on how to get there using the definition
I don't know how to get out of this infinite integration of sin(t) and cos(t)
what I've been trying:
$\begin{aligned} & \text { g) } f(t)=e^{-t} \operatorname{sen}(t) \\ & \mathcal{L }\left\{e^{-t} \operatorname{sen}(t)\right\}=\int_0^{\infty} e^{-s t} \cdot e^{-t} \operatorname{sen}(t) d t=\int_0^{\infty} e^{-t(s+1)} \operatorname{sen}(t)dt \\ \end{aligned}$
$\begin{aligned} & \left.\left(-\frac{\operatorname{sen}(t)}{s+1}\right) e^{-t(s+1)}\right)_0^{\infty}+\frac{1}{s+1} \int e^{-t(s+1)} \cos (t) dt \\ & \frac{1}{s+1}\left[-\frac{\cos (t)}{s+1} e^{-t(s+1)}-\frac{1}{s+1} \int e^{-t(s+1)} \operatorname{sen}(t) d t\right] \end{aligned}$