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To solve complex rational functions like \( F(s)= \frac{s+1}{s(s^2+s+1)} \), we often use **Partial Fraction Decomposition**. This technique allows us to break down a complicated fraction into simpler fractions that are easier to handle. Essentially, we want to express the given function as a sum of simpler fractions.This process begins by representing the function in a decomposed form:\[ F(s) = \frac{s+1}{s(s^2+s+1)} = \frac{A}{s} + \frac{Bs+C}{s^2+s+1} \]Here, \(A\), \(B\), and \(C\) are coefficients that we need to find. By equating the numerators on both sides, we get a new equation that can be simplified by expanding and combining like terms. This allows us to solve for the coefficients step-by-step.

With the partial fraction form set up, the next step is to find the **Laplace Transform Coefficients**. These coefficients, \( A, B, \) and \( C \), are crucial for expressing the original function in a decomposed form.We start by rewriting the equation:\[ s+1 = A(s^2+s+1) + (Bs+C)s \]Expanding and combining like terms, we get:\[ s+1 = As^2 + As + A + Bs^2 + Cs = (A+B)s^2 + (A+C)s + A \]By comparing the coefficients on both sides for \( s^2 \), \( s \), and the constant term, we get the following equations:

• For \( s^2 \): \( A + B = 0 \)
• For \( s \): \( A + C = 1 \)
• For the constant term: \( A = 1 \)

Solving these equations, we find:

• \( A = 1 \)
• \( B = -1 \)
• \( C = 0 \)

Therefore, the partial fractions are:\[ F(s) = \frac{1}{s} + \frac{-s}{s^2+s+1} \].

The final goal is to find the **Inverse Laplace Transform** of the decomposed fractions. We handle each term separately:1. **First term**: \( \frac{1}{s} \)The Inverse Laplace Transform is straightforward for \( \frac{1}{s} \), which gives us:\[ L^{-1} \{ \frac{1}{s} \} = 1 \].2. **Second term**: \( \frac{-s}{s^2+s+1} \)For this term, we need to simplify the denominator into a recognizable form. By completing the square, we transform it:\[ \frac{-s}{(s + 1/2)^2 + 3/4} \]This corresponds to a known inverse Laplace transform form:\[ \frac{s-a}{(s-a)^2 + b^2} \rightarrow e^{at} \cos (bt) \].Hence, the second term translates to:\[ L^{-1} \{ \frac{-s}{s^2+s+1} \} = - e^{-\frac{t}{2}} \cos \left( \frac{\sqrt{3}}{2} t \right) \].Finally, we combine the results:\[ L^{-1} \{ \frac{s+1}{s(s^2+s+1)} \} = 1 - e^{-\frac{t}{2}} \cos \left( \frac{\sqrt{3}}{2} t \right) \].Understanding these steps allows students to tackle similar problems with confidence.