Given the Laplace transform $F\left(s\right)=\frac{1}{s^2+1}$, find the corresponding function $f\left(t\right)$.

• A. $e^t$
• B. $\cos t$
• C. $\sin t$
• D. $t$

Answer: (C)

Verified step by step guidance

1. Step 1: Recall the definition of the Laplace transform and its inverse. The Laplace transform maps a function f(t) in the time domain to F(s) in the frequency domain. To find the corresponding function f(t), we need to apply the inverse Laplace transform to F(s).
2. Step 2: Identify the form of F(s). The given Laplace transform is F(s) = 1 / (s^2 + 1). This matches the standard Laplace transform formula for sin(t), which is 1 / (s^2 + ω^2) where ω = 1.
3. Step 3: Use the inverse Laplace transform formula. The inverse Laplace transform of 1 / (s^2 + ω^2) is sin(ωt). In this case, ω = 1, so the corresponding function is sin(t).
4. Step 4: Verify the result by recalling the Laplace transform of sin(t). The Laplace transform of sin(t) is 1 / (s^2 + 1), which matches the given F(s). This confirms that the inverse Laplace transform of F(s) is indeed f(t) = sin(t).
5. Step 5: Conclude that the corresponding function f(t) is sin(t). This is the result obtained by applying the inverse Laplace transform to F(s).