5.32 how to use Laplace transform?

restart; 
f:= t->sin(omega*t) ; 
L:=convert(inttrans[laplace](f(t),t,s),int);
 

\[ {\frac {\omega }{{\omega }^{2}+{s}^{2}}} \]

To find the inverse, do:

 inttrans[invlaplace](L,s,t);
 

\[ \sin \left ( \omega \,t \right ) \]

For unit step, use

_EnvUseHeavisideAsUnitStep:=true; 
f:=Heaviside(t-a); 
INV:=inttrans:-laplace(f,t,s) assuming a>0;  #make sure to use a>0
 

\[ \frac {{\mathrm e}^{-s a}}{s} \]

Another example

_EnvUseHeavisideAsUnitStep:=true; 
f:=Heaviside(t)-Heaviside(t-a); 
INV:=inttrans:-laplace(f,t,s) assuming a>0;  #make sure to use a>0
 

\[ \frac {1-{\mathrm e}^{-s a}}{s} \]