Find the Laplace transform of the following ft 2 3t e21 5

Find the Laplace transform of the following. f(t) = [2 + 3t e^-21 + 5e^-10t - 3cos(5t) + 3sin(5t)] u(t)

Solution

Here we know that thethe Laplace transform of some standard signals are

x(t)

1

e-^at

te-^at

sin(at)

cos(at)

X(s)

1/s

1/(s+a)

1/(s+a)²

a/(s²+ a²)

S/(s²+ a²)

form the given problem

f(t) = [2 + 3te^-2t + 5 e^-10t - 3 cos(5t) + 3 sin(5t) ] u(t)

the the laplace transform of the above function is

F(s) = L( 2 (u(t))) + L(3te^-2t (u(t))) + L(5 e^-10t u(t)) - L(3 cos(5t)u(t) ) + L(3 sin(5t)u(t))

= 2/s + 3/(s+2)² + 5/(s+10) - 3x s/(s² + 25) + 3 X 5/ (s² + 25)

= 2/s +   3/(s+2)² + 5/(s+10) - 3s/(s² + 25) + 15/ (s² + 25)

= 2/s +   3/(s+2)² + 5/(s+10) + 15-3s/(s² + 25)

x(t) 1 e-at te-at sin(at) cos(at)

X(s) 1/s 1/(s+a) 1/(s+a)2 a/(s2+ a2) S/(s2+ a2)