B-2-11. find the inverse laplace transform of s + 1/s(s^2 + s +1)

Step-by-step explanation:

let's start by separating the fraction into two new smaller fractions

.

First, s(s^2+s+1) must be factorized the most, and it is already. Every factor will become the denominator of a new fraction.

Where A, B and C are unknown constants. The numerator of s is a constant A, because s is linear, the numerator of s^2+s+1 is a linear expression Bs+C because s^2+s+1 is a quadratic expression.

Multiply both sides by the complete denominator:

Simplify, reorganize and compare every coefficient both sides:

Solving the system, we find A=1, B=-1, C=0. Now:

Then, we can solve the inverse Laplace transform with simplified expressions:

The first inverse Laplace transform has the formula:

For:

We have the formulas:

We have to factorize the denominator:

It means that:

So a=-1/2 and b=(√3)/2. Then:

Finally:

step-by-step explanation: 2