Page 14 - LAPLACE TRANSFORM

Example 5

Find Laplace Transform, F(s) by using Definition Method, when

f ( t) = cos at

F s( ) = ∫ ∞ e −st cos at dt …………………………………(1)
Ο

# Using part by part integration
u = e − st du = − se −st dt

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∫
∫ dv = cos at dt v = a sin at

e −st 1
(
−
st
F s( ) = sin at − ∫ sin at − se ) dt
a a
e − st s
F s( ) sinat + ∫ e − st sinat dt …………………………….(2)
a a

From (2), let,

A = ∫ e − st sinat dt ………………………………………………… (3)

e − st s
∴ F s( ) = sinat + A ………………………………………………... (4)
a a

From (3) integrate part by part,

u = e − st du = − se − st dt

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