Laplace transform of e^{θt} sin (ωt) is:
Concept:
Bilateral Laplace transform:
\(L\left[ {x\left( t \right)} \right] = x\left( s \right) = \;\mathop \smallint \limits_{  \infty }^\infty x\left( t \right){e^{  st}}dt\)
Unilateral Laplace transform:
\(L\left[ {x\left( t \right)} \right] = x\left( s \right) = \;\mathop \smallint \limits_0^\infty x\left( t \right){e^{  st}}dt\)
Frequency shifting property:
If X(s) is the Laplace transform of x(t), then
\({e^{at}}X\left( s \right) \leftrightarrow X\left( {s  a} \right)\)
Calculation:
f(t) = eθt sin (ωt)
\(L[sin \ ω_0t]= \frac{\omega}{{{s^2} + \omega^2}}\)
By using frequency shifting property,
The Laplace transform of f(t) is
\(F\left( s \right) = \frac{\omega}{(s\theta)^2+ω^2 }\)
Some important Laplace transforms:

f(t) 
f(s) 
ROC 
1. 
δ(t) 
1 
Entire splane 
2. 
eat u(t) 
\(\frac{1}{{s + a}}\) 
s >  a 
3. 
eat u(t) 
\(\frac{1}{{s + a}}\) 
s <  a 
4. 
cos ω0 t u(t) 
\(\frac{s}{{{s^2} + \omega _0^2}}\) 
s > 0 
5. 
teat u(t) 
\(\frac{1}{{{{\left( {s + a} \right)}^2}}}\) 
s >  a 
6. 
sin ω0t u(t) 
\(\frac{{{\omega _0}}}{{{s^2} + \omega _0^2}}\) 
s > 0 
7. 
u(t) 
1/s 
s > 0 