Question

Use Definition 7.1.1, DEFINITION 7.1.1 Laplace Transform Let f be a function defined for t ≥ 0. Then the integral ℒ{f(t)} = [infinity] e−stf(t) dt 0 is said to be the Laplace transform of f, provided that the integral converges. to find ℒ{f(t)}. f(t) = cos t, 0 ≤ t < π 0, t ≥ π

Answer 1

the Laplace transform of f(t) , ℒ{f(t)](s) = 0

Explanation:

the Laplace transform for the function f(t)=cos t, 0 ≤ t < π and 0, t ≥ π is

ℒ{f(t)](s) =∫ f(t) *e^(-s*t) dt from 0 to ∞ = ∫ cos t * e^(-s*t) dt from 0 to π

then

ℒ{f(t)](s) = ∫cos (t) * e^(-s*t) dt = sin(t)*e^(-s*t) /₀π - ∫ cos(t)* (-s)e^(-s*t) dt = 0 +s*∫ cos(t)* e^(-s*t) dt

thus

∫cos (t) * e^(-s*t) dt - s*∫ cos(t)* e^(-s*t) dt = 0

(1-s)* ∫ cos(t)* e^(-s*t) dt = 0

therefore since s can be different from 0

∫ cos(t)* e^(-s*t) dt = 0 and thus ℒ{f(t)](s) = 0