Final answer:
C. x(t)=8(e²t ) 2 +6cos(4t)+8sin(4t) In this case, the expression (s−2)²/8 + s² + 166s + 8 is initially given in the Laplace domain.
Explanation:
The inverse Laplace transform of X(s) = (s−2)²/8 + s² + 166s + 8 results in x(t) = 8(e²t )² + 6cos(4t) + 8sin(4t). This solution corresponds to option C. The function contains exponential terms, a cosine function, and a sine function. The coefficient of the exponential term matches the expression derived from the inverse Laplace transform, distinguishing it from the other options. The cosine and sine terms with coefficients 6 and 8, respectively, are also consistent with the original function's structure.
The inverse Laplace transform involves decomposing a function in the complex domain back into its original time-domain representation. In this case, the expression (s−2)²/8 + s² + 166s + 8 is initially given in the Laplace domain.
The resulting time-domain function comprises exponential terms and trigonometric functions. The presence of e²t squared is a crucial factor that separates this solution from others, aligning with the given expression. The cosine and sine terms with their respective coefficients further confirm the correctness of this inverse Laplace transform. Therefore, option C accurately represents the time-domain function corresponding to the given Laplace expression.
This method efficiently reverses the transformation from the Laplace domain to the time domain, showcasing the nature of the original function with its exponential and trigonometric components.