Step by step
The question
Question: find the Laplace transform of cos(2t).
ℒ{cos(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.
cos(bt) → ss² + b²
2 Use the table
Each part by its row: cos(2t) gives s/(s² + 4).
ℒ{cos(2t)} = ss2 + 4
3 Check
Let's check: read the table backwards.
s/(s² + 4) comes from cos(2t): the question.
Right!
backwards: ss2 + 4 → cos(2t)
Let's look back.
First: the idea.
Then: use the table.
Last: check.
Answer
ℒ{cos(2t)} = ss2 + 4