Final answer:
To obtain the inverse Laplace transform of F(s) = 2/s(s+1)(s^2+2s+2), use partial fraction decomposition, solve for the constants, and then apply the inverse Laplace transform to each term using known transform pairs and properties.
Step-by-step explanation:
The inverse Laplace transform of the function F(s) = 2 / s(s+1)(s^2 + 2s + 2) can be found using partial fraction decomposition. Considering the factors in the denominator, you would express F(s) as a sum of simpler fractions, where each term corresponds to a factor in the denominator as follows:
F(s) = A/s + B/(s+1) + (Cs + D)/(s^2 + 2s + 2)
After solving for the constants A, B, C, and D, you apply the inverse Laplace transform to each term separately using known Laplace transform pairs and properties such as the first and second shift theorems. For example, the term with s^2 + 2s + 2 in the denominator will end up involving complex numbers due to the fact that this corresponds to a complex pair of poles. The actual inverse Laplace transform can get quite intricate and requires familiarity with the method of complex inversion or using standard Laplace pairs to find the time-domain function.