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We are asked to find the inverse Laplace transform of the function:

\[
F(s) = \frac{3s^2 + 1}{s^2(s^2 + 1)}
\]

### Step 1: Decompose the function
We start by breaking down \( F(s) \) into simpler fractions using partial fraction decomposition.

\[
F(s) = \frac{3s^2 + 1}{s^2(s^2 + 1)}
\]

Let’s rewrite this as:

\[
F(s) = \frac{A}{s} + \frac{B}{s^2} + \frac{Cs + D}{s^2 + 1}
\]

Multiplying both sides by \( s^2(s^2 + 1) \) to clear the denominators:

\[
3s^2 + 1 = A s(s^2 + 1) + B(s^2 + 1) + (Cs + D)s^2
\]

Expanding the right-hand side:

\[
3s^2 + 1 = A s^3 + A s + B s^2 + B + C s^3 + D s^2
\]

Group terms by powers of \( s \):

\[
3s^2 + 1 = (A + C) s^3 + (B + D) s^2 + A s + B
\]

### Step 2: Match coefficients
We can now match the coefficients of \( s^3 \), \( s^2 \), \( s \), and the constant:

- Coefficient of \( s^3 \): \( A + C = 0 \) ⟹ \( A = -C \)
- Coefficient of \( s^2 \): \( B + D = 3 \)
- Coefficient of \( s \): \( A = 0 \) ⟹ \( A = 0 \) ⟹ \( C = 0 \)
- Constant term: \( B = 1 \)

From \( A = 0 \), \( C = 0 \), \( B = 1 \), and \( D = 3 \), the partial fraction decomposition becomes:

\[
F(s) = \frac{1}{s^2} + \frac{3}{s^2 + 1}
\]

### Step 3: Find the inverse Laplace transforms
- The inverse Laplace transform of \( \frac{1}{s^2} \) is \( t \).
- The inverse Laplace transform of \( \frac{3}{s^2 + 1} \) is \( 3 \sin(t) \).

### Step 4: Combine the results
The inverse Laplace transform of \( F(s) \) is:

\[
f(t) = t + 3 \sin(t)
\]

### Step 5: Match with the given options
This matches option **(a)**: \( 2 \sin(t) + t \).

### Final Answer:
The correct answer is **(a)**.

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Would you like further details or explanations? Here are some related questions to expand your understanding:

1. How is partial fraction decomposition useful in inverse Laplace transforms?
2. What is the inverse Laplace transform of \( \frac{1}{s(s^2 + 1)} \)?
3. How does Laplace transform simplify solving differential equations?
4. How can you identify when to use inverse Laplace transforms in a problem?
5. What is the general approach to solving Laplace transforms involving polynomials?

**Tip:** Always ensure you match terms carefully when using partial fractions to avoid errors in coefficients!

The inverse Laplace transform of F(s) = (3s^2 + 1) / (s^2 * (s^2 + 1))

Learn how to find the inverse Laplace transform of F(s) = (3s^2 + 1) / (s^2 * (s^2 + 1)) using partial fraction decomposition. This step-by-step solution includes the breakdown of partial fractions and the application of inverse Laplace transform formulas. Suitable for undergraduate students studying differential equations and transforms.

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