Page 15 - LAPLACE TRANSFORM

1
∫
∫
dv = sin at dt v = − cos at
a

e − st 1
( se
A = − cos at − ∫ − cos at − − st )dt
a a

e − st s
A −= cos at − ∫ e − st cos at dt ………………………………..(5)
a a

From (1) and (5) ,

e −st s
A = − cos at − F ( ) s ………………………………………….(6)
a a

From (6) we substitute into (4)

e −st  s e −st s 
F s( ) = sin at +  − cos at − F s
( )
a a  a a 
e − st se −st s 2
= − cos at − F ( ) s
a a 2 a 2

2
s F ( ) s e − st se −st
= F s( ) + − cos at
a 2 a a 2

2
 a + s  e − st  s 
2
=  F ( ) s = sin at − cos at
 a 2  a  a 

ae − st  s  ∞
= F s( ) = sin at − cos at
2
2
s + a  a  ο

a  1  s 
(
F s) =  −  0 − cos 0
s + a  ∞  a 
2
2
a  s 
=  
2
2
s + a  
a

s
=
s + a 2
2

8