Step by step
The question
Question: find the inverse Laplace transform of 5/s³.
ℒ⁻¹{5s3} =
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.
tⁿ needs n!
on top.
The extra number waits in front.
ℒ⁻¹ = 52 × 2!s3
tn → n!sn + 1
2 Use the table backwards
Careful!
2!
= 2 belongs on top.
So the number in front is 5/2.
Careful: not 5/s³ → 5t² ✗
Read the table backwards, piece by piece: 5/2 × t².
f(t) = 52t2
3 Check
Let's check: the table forwards turns our answer back into 5/s³.
Right!
forwards: 52t2 → 5s3
Let's look back.
First: make it fit the table.
Then: use the table backwards.
Last: check.
Answer
f(t) = 52t2