Page 12 - LAPLACE TRANSFORM

Find Laplace Transform, of the following by using the Definition

of Laplace Transform.

f ( t) = sin at

∞

F s( ) = ∫ ο e −st sinat dt ..........(1)

#Formula for part by part integration

=
∫ udv uv − ∫ vdu
Integrate part by part;

u = e − st du = − se − st dt

1
∫
∫
dv = sin at dt v = − cos at
a
∴F s( ) = − e − st cos at − − 1 cos at ( se− − st )dt

a ∫ a

−= e − st cos at − s ∫ e −st cos at dt …………………………..(2)

a a

From (2) let:

A = e −st cos at dt
∫

− st − st

u = e du = − se dt

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

5