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)} = ∞ 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

The Laplace Transform of the piecewise Function is .

Given that Function is continuous and integrable. The Laplace Transform is defined by the following expression:

, (1)

We have the following piecewise Function:

(2)

By (1) and using Integral Properties, the Laplace Transform of the piecewise Function is:

(3)

By Part Integration, we have the following outcome:

, , ,

(4)

The Laplace Transform of the piecewise Function is .

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The Laplace transform is then

Let denote the integral we want to compute. Integrating by parts, setting

gives

Integrate by parts again, setting

Then