Use Definition 7.1.1, DEFINITION 7.1.1 Laplace Transform LetUse 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)}. (Write your answer as a function of s.) WebAssign Plotf 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)}. (Write your answer as a function of s.) f(t) = cos(t), 0 ≤ t < π 0, t ≥ π

Step-by-step explanation:

Using the formula for Laplace the transformations if  is the converted function then

To solve that integral you need to use integration by parts, when you do integration by parts you get that

.

The laplace transform is

Step-by-step explanation:

Let us asume that f(t) =0 for t<0. So, by definition, the laplace transform is given by:

To solve this integral, we will use integration by parts. Let u= cos(t)  and dv = , so v= and du = -sin(t), then, in one step of the integration we have that

Let . We will integrate I_2 again by parts. Choose u = sin(t) and dv = . So

Therefore,

which is an equation for the variabl I. Solving for I we have that

Then,

.

Note that since the sine function is 0 at 0 and pi, we must only care on the first term. Then