Inverse Laplace Transform of a Rational Function

Find the inverse Laplace transform of $\frac{3s + 8}{s^2 + 2s + 5}$

The inverse Laplace transform is a powerful tool in differential equations and systems analysis, often used to reconstruct time-domain signals from their s-domain representations. In tackling the problem of finding the inverse Laplace transform of a given function, one needs to be familiar with techniques such as partial fraction decomposition or completing the square, both of which can simplify the process when dealing with complex rational functions or polynomials.

Understanding how to manipulate algebraic expressions in the Laplace domain is crucial. This involves recognizing standard forms and possibly making algebraic manipulations to match standard Laplace transform pairs. For instance, when faced with a quadratic in the denominator, as in this problem, completing the square can be an effective strategy to rewrite the function in a form that matches a standard inverse Laplace transform pair.

The process not only involves algebraic manipulation but also a strong grasp of the theoretical foundation. The conversion from the s-domain to the time domain theoretically underpins much of control theory and systems engineering, emphasizing the importance of the inverse Laplace transform in engineering and physics.