Question

102–106. Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by F(s) = ∫[0 to ∞] e^(-st) f(t) dt, where we assume s is a positive real number. For example, to find the Laplace transform of f(t) = e^(-t), the following improper integral is evaluated using integration by parts:
F(s) = ∫[0 to ∞] e^(-st) e^(-t) dt = ∫[0 to ∞] e^(-(s+1)t) dt = 1/(s+1).
Verify the following Laplace transforms, where a is a real number.
106. f(t) = cos(at) → F(s) = s/(s² + a²)

Accepted Answer

Start with the definition of the Laplace transform for the function $$f(t) = \cos(at)$$:

$$F(s) = \int_0^{\infty} e^{-st} \cos(at) \, dt$$ Recall that the integral of the product of an exponential and a cosine function can be evaluated using the formula:

$$\int_0^{\infty} e^{-pt} \cos(qt) \, dt = \frac{p}{p^2 + q^2}$$  
where $$p \u003e 0$$ and $$q$$ are real numbers. In this problem, identify $$p = s$$ and $$q = a$$, since the integral matches the form with $$e^{-st}$$ and $$\cos(at). $$Apply the formula directly to get:

$$F(s) = \frac{s}{s^2 + a^2}$$  
which is the Laplace transform of $$\cos(at). To $$fully verify, you could also perform the integral by expressing $$\cos(at) as $$the real part of $$e^{iat}$$ and then integrating $$e^{-(s - ia)t}$$, ensuring the integral converges for $$s \u003e 0.$$

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Key Concepts

Definition of the Laplace Transform

Improper Integrals and Convergence

Laplace Transform of Trigonometric Functions