USEFUL FORMULAS: ( YES YOU MAY REMOVE THIS PAGE )
Laplace Transforms:
f (t)
F (s)
1
1
s, s > 0
1
t
s2 , s > 0
n!
n
t , n = 1, 2, 3 sn+1
,s > 0
1
at
e
s−a , s > a
s
cos(ωt)
s2 +ω 2 , s > 0
ω
sin(ωt)
s2 +ω 2 , s > 0
e−as
u(t − a)
s
δ(t − a)
e−as
L {f 0 (t)} = sF (s) − f (0)
L {tn f (t)} = (−1)n F (n) (s)
Z ∞
f (t)
L
=
F (σ)dσ
t
s
L eat f (t) = F (s − a)
L {f (t − a)u(t − a)} = e−as F (s)
Z t
L
f (τ )g(t − τ )dτ = F (s)G(s)
0
Z t
F (s)
L
f (τ )dτ =
s
0
Useful Integrals:
Z
dx
p
p
= sinh−1 (x) + C = ln(x + (1 + x2 )) + C
(1 + x2 )
Z
dx
p
= sin−1 (x) + C
2
(1 − x )
Z
dx
= tan−1 (x) + C
2
1+x
Z
tan(x) = − ln |cos(x)| + C
(1)
Trigometric Identities:
sin(a + b) = sin(a) cos(b) + cos(a) sin(b)
sin(a − b) = sin(a) cos(b) − cos(a) sin(b)
cos(a + b) = cos(a) cos(b) − sin(a) sin(b)
cos(a − b) = cos(a) cos(b) + sin(a) sin(b)
(2)
1