Question

# use properties of the Laplace transform and the table of Laplace transforms to determine L[f] f(t)=e^{3t}cos5t-e^{-t}sin2t

Laplace transform
use properties of the Laplace transform and the table of Laplace transforms to determine L[f] $$f(t)=e^{3t}\cos5t-e^{-t}\sin2t$$

Step 1 Given function is $$f(t)=e^{3t}\cos5t-e^{-t}\sin2t$$ Step 2 Obtain the Laplace transform. $$L\left\{e^{at}\cos bt\right\}=\frac{s-a}{(s-a)^2+b^2}$$
$$L\left\{e^{at}\sin bt\right\}=\frac{b}{(s-a)^2+b^2}$$
$$L\left\{f(t)\right\}=L\left\{e^{3t}\cos 5t -e^{-t}\sin 2t\right\}$$
$$=L\left\{e^{3t}\cos 5t\right\}-L\left\{e^{-t}\sin 2t\right\}$$
$$=\frac{s-3}{(s-3)^2+5^2}-\frac{2}{(s-(-1))^2+2^2}$$
$$=\frac{s-3}{(s-3)^2+25}-\frac{2}{(s+1)^2+4}$$