Step by step
The question
Question: find the Laplace transform of te^(2t).
ℒ{te^(2t)} =
1 The idea
The Laplace transform turns a function of t into a function of s.
It is an integral, but we almost never work it out: we use a table made from it.
ℒ{f(t)} = ∫ from 0 to ∞ of e^(−st) f(t) dt
The table rows we need.
tne^(at) → n!(s − a)n + 1
2 Use the table
Each part by its row: te^(2t) gives 1/(s − 2)².
ℒ{te^(2t)} = 1(s − 2)2
3 Check
Let's check: read the table backwards.
1/(s − 2)² comes from te^(2t): the question.
Right!
backwards: 1(s − 2)2 → te^(2t)
Let's look back.
First: the idea.
Then: use the table.
Last: check.
Answer
ℒ{te^(2t)} = 1(s − 2)2