Which of the following is the correct inverse Laplace Transform of
𝑋
(
𝑠
)
=
2
𝑠
+
1
𝑠
2
+
2
𝑠
+
5
?

Question

Which of the following is the correct inverse Laplace Transform of 
𝑋
(
𝑠
)
=
2
𝑠
+
1
𝑠
2
+
2
𝑠
+
5
?

BlackTom AI · Accepted Answer

{
  "reasoning": "To tackle this inverse Laplace transform, I’ll break X(s) into parts whose time-domain forms we know.

First, consider the term 2/s. Its inverse Laplace transform is simply the constant 2 for t ≥ 0, i.e., L^{-1}{2/s} = 2. This contributes a constant component 2·u(t) in the time domain.

Next, examine 1/(s^2 + 2s + 5). Completing the square gives s^2 + 2s + 5 = (s + 1)^2 + 4. This matches the standard form for e^{-at} sin(bt) and e^{-at} cos(bt) terms:
- L{e^{-t} cos(2t)} = (s + 1)/[(s + 1)^2 + 4]
- L{e^{-t} sin(2t)} = 2/[(s + 1)^2 + 4]

We can express 1/[(s + 1)^2 + 4] as (1/2) · [ 2/((s + 1)^2 + 4) ], which corresponds to (1/2)·L{e^{-t} sin(2t)}. Therefore, L^{-1}{1/[(s + 1)^2 + 4]} = (1/2) e^{-t} sin(2t).

Putting the pieces together, the inverse Laplace transform of X(s) is:
x(t) = 2 + (1/2) e^{-t} sin(2t), for t ≥ 0 (often written as x(t) = 2 + (1/2) e^{-t} sin(2t)·u(t)).

Now let’s evaluate each provided option against this result, noting their structure and any missing pieces.

Option 1: e^{−t}[ (1/2) cos(2t) − (1/2) sin(2t) ] u(t)
- Why this is not correct: This form corresponds to a linear combination of e^{-t} cos(2t) and e^{-t} sin(2t). If you take its Laplace transform, you would get expressions involving (s+1) factors in the numerator that do not recreate the separate 2/s term and the (s+1)^2+4 denominator form we started with. Specifically, it lacks the pure constant 2/u(t) term and mixes the decaying oscillatory components in a way that does not match X(s).

Option 2: e^{−t}[ 2 cos(2t) − (1/2) sin(2t) ] u(t)
- Why this is not correct: This is a decaying exponential times a combination of cos and sin with particular coefficients. Its Laplace transform would be a combination of (s+1) terms divided by ((s+1)^2+4), not a sum of 2/s and 1/((s+1)^2+4). It fails to produce the separate constant 2/s piece and the proper (s+1) structure needed for the second term.

Option 3: e^{−2t}[ 2 cos(t) + (1/2) sin(t) ] u(t)
- Why this is not correct: Here the exponential decay rate is e^{−2t} with a base frequency of t, not matching the required e^{−t} and frequency 2t in the original components. The mismatch in the exponential factor and frequency means its Laplace transform will not reconstruct X(s).

Option 4: e^{−t}[ 2 cos(2t) + (1/2) sin(2t) ] u(t)
- Why this is not correct: This option has the same decay rate e^{−t} and the same oscillatory terms as Option 2 but with a plus sign on the sin term. Even if the sign were adjusted, the overall form still represents a decaying oscillation, not the sum of a constant 2 and a separate (1/2) e^{−t} sin(2t) term. It does not produce the 2/s component.

Option 5: e^{−2t}[ 2 cos(2t) − (1/2) sin(2t) ] u(t)
- Why this is not correct: This option uses e^{−2t} instead of e^{−t}, and the frequency inside the cosine is 2t rather than the needed 2t inside the sin term for the (1/2) e^{−t} sin(2t) part. This combination does not align with the (s+1)^2+4 denominator structure after inversion and thus cannot equal X(s).

Option 6: e^{−t}[ 1/2 cos(2t) − 2 sin(2t) ] u(t)
- Why this is not correct: Similar issues persist: the coefficients inside the brackets correspond to a specific combination of e^{−t} cos(2t) and e^{−t} sin(2t), which would yield a Laplace transform with numerator forms that do not decompose into 2/s plus 1/[(s+1)^2+4]. It also misses the pure constant term 2 present in the time domain.

Summary of the comparison: none of the given options reproduce the correct time-domain expression x(t) = 2 + (1/2) e^{−t} sin(2t)·u(t). The only way to obtain the constant 2 is from the 2/s term in X(s); all the listed options express purely decaying oscillatory behavior multiplied by u(t), without a separate constant offset.”}

ECEN-314:200,501 Final Exam- Requires Respondus LockDown Browser

Which of the following is the correct inverse Laplace Transform of 𝑋 ( 𝑠 ) = 2 𝑠 + 1 𝑠 2 + 2 𝑠 + 5 ?

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