Final answer:
To find the Laplace transform of the given function, f(t) = eat sinh bt, use the properties of the Laplace transform and the definition of the sinh function. The Laplace transform of f(t) = eat sinh bt is F(s) = b / (s - a) / ((s - a)^2 + b^2).
Step-by-step explanation:
The Laplace transform of the given function f(t) = eat sinh bt can be found using the properties of the Laplace transform and the definition of the sinh function.
Let's start by using the definition of the Laplace transform:
L{f(t)} = F(s) = ∫[t=0][∞]f(t)e^-st dt
By substituting the given function and applying the properties of the Laplace transform, we can simplify and evaluate the integral to find the Laplace transform of f(t).
The Laplace transform of f(t) = eat sinh bt is F(s) = b / (s - a) / ((s - a)^2 + b^2).