Laplace transform converts a time domain function to sdomain function by integration from zero to infinity
of the time domain function, multiplied by e^{st}.
The Laplace transform is used to quickly find solutions for differential equations and integrals.
Derivation in the time domain is transformed to multiplication by s in the sdomain.
Integration in the time domain is transformed to division by s in the sdomain.
The Laplace transform is defined with the L{} operator:
The inverse Laplace transform can be calculated directly.
Usually the inverse transform is given from the transforms table.
Function name  Time domain function  Laplace transform 

f (t) 
F(s) = L{f (t)} 

Constant  1  
Linear  t  
Power  t^{ n} 

Power  t^{ a} 
Γ(a+1) ⋅ s ^{(a+1)} 
Exponent  e^{ at} 

Sine  sin at 

Cosine  cos at 

Hyperbolic sine 
sinh at 

Hyperbolic cosine 
cosh at 

Growing sine 
t sin at 

Growing cosine 
t cos at 

Decaying sine 
e^{ at }sin ωt 

Decaying cosine 
e^{ at }cos ωt 

Delta function 
δ(t) 
1 
Delayed delta 
δ(ta) 
e^{as} 
Property name  Time domain function  Laplace transform  Comment 

f (t) 
F(s) 

Linearity  a f (t)+bg(t)  aF(s) + bG(s)  a,b are constant 
Scale change  f (at)  a>0  
Shift  e^{at} f (t)  F(s + a)  
Delay  f (ta)  e^{as}F(s)  
Derivation  sF(s)  f (0)  
Nth derivation  s^{n}f (s)  s^{n1}f (0)  s^{n2}f '(0)...f^{ (n1)}(0)  
Power  t^{ n} f (t)  
Integration  
Reciprocal  
Convolution  f (t) * g (t)  F(s) ⋅ G(s)  * is the convolution operator 
Periodic function  f (t) = f (t+T) 
Find the transform of f(t):
f (t) = 3t + 2t^{2}
Solution:
ℒ{t} = 1/s^{2}
ℒ{t^{2}} = 2/s^{3}
F(s) = ℒ{f (t)} = ℒ{3t + 2t^{2}} = 3ℒ{t} + 2ℒ{t^{2}} = 3/s^{2} + 4/s^{3}
Find the inverse transform of F(s):
F(s) = 3 / (s^{2} + s  6)
Solution:
In order to find the inverse transform, we need to change the s domain function to a simpler form:
F(s) = 3 / (s^{2} + s  6) = 3 / [(s2)(s+3)] = a / (s2) + b / (s+3)
[a(s+3) + b(s2)] / [(s2)(s+3)] = 3 / [(s2)(s+3)]
a(s+3) + b(s2) = 3
To find a and b, we get 2 equations  one of the s coefficients and second of the rest:
(a+b)s + 3a2b = 3
a+b = 0 , 3a2b = 3
a = 3/5 , b = 3/5
F(s) = 3 / 5(s2)  3 / 5(s+3)
Now F(s) can be transformed easily by using the transforms table for exponent function:
f (t) = 3e^{2t}  3e^{3t}