TABLE OF LAPLACE TRANSFORM
1. L1 = 1s
L −1  1s  = 1
2. Lt n  = n!
s n+1
t n−1
L −1  1n  =
s
n − 1!
3. Le at  = s −1 a
L −1  s −1 a  = e at
4. Lsinat =
a
s2 + a2
L −1 
1  = 1 sinat
a
s2 + a2
5. Lcosat =
s
s + a2
L −1 
s  = cosat
s + a2
6. Lsinhat =
a
s2 − a2
L −1 
1  = 1 sinhat
a
s2 − a2
7. Lcoshat =
s
s2 − a2
L −1 
2
−as
2
s  = coshat
s2 − a2
−as
8. Lu a t = e s
L −1  e s  = u a t , a > 0.
First Differentiation Formula
Let Xs be the Laplace transform of xt, that is Xs = Lxt and let D = d .
dt
n
n
n−1
n−2 ′
n−1
LD x = s X − s x0 − s x 0 −⋅ ⋅ ⋅ −x
0 , n = 1, 2, 3, ⋅ ⋅ ⋅
In Particular if n = 1 , Lx ′  = s X − x0 , and if n = 2 , Lx ′′  = s 2 X − s x0 − x ′ 0.
In the following formulae, Fs = Lft, so ft = L −1 Fs, and a > 0.
Second Differentiation Formula
n
Lt n ft = −1 n d n Lft
ds
, n = 1, 2, 3, ⋅ ⋅ ⋅
The First Shift Formula
Inverse Version
Le at ft = Fs − a OR
L −1 Fs = e at L −1 Fs + a OR
Le at ft = Lft| s→ s−a
L −1 Fs = e at L −1 Fs| s→ s+a 
The Second Shift Formula
Inverse Version
Lu a tgt = e −as Lgt + a OR
L −1 e −as Fs = u a tft − a
Lu a tgt = e −as Lgt| t→ t+a 
L −1 e −as Fs = u a tL −1 Fs| t→ t−a
Completing The Squares:
If A, B, and C are real numbers and if A > 0 then,
As 2 + Bs + C =  A s +
2
B
 2 + C − B4A 
2 A
OR