Step by step
The question
Question: find the inverse Laplace transform of 1/(s² + 4).
ℒ⁻¹{1s2 + 4} =
The inverse transform goes back: which function of t has this transform?
We read the table backwards.
ℒ⁻¹: back from s to t
1 Make it fit the table
Make each piece look exactly like a row of the table.
sin needs b on top.
The extra number waits in front.
ℒ⁻¹ = 12 × 2s2 + 4
sin(bt) → bs² + b²
2 Use the table backwards
Careful!
sin(2t) has 2 on top.
With 1 on top we get 1/2 sin(2t).
Careful: not 1/(s² + 4) → 1sin(2t) ✗
Read the table backwards, piece by piece: ½sin(2t).
f(t) = 12sin(2t)
3 Check
Let's check: the table forwards turns our answer back into 1/(s² + 4).
Right!
forwards: 12sin(2t) → 1s2 + 4
Let's look back.
First: make it fit the table.
Then: use the table backwards.
Last: check.
Answer
f(t) = 12sin(2t)