Laplace transform multiplication by t: How to find t cos 3t

Laplace transform of a multiplication by t of cosine 3t.Step by step of how to find multiplication by t of t cos 3t.
Laplace transform is the integral transform of the given derivative function with real variable t to convert into a complex function with variable s. For t ≥ 0, let f(t) be given and assume the function satisfies certain conditions to be stated later on.
The Laplace transform of f(t), that is denoted by L{f(t)} or F(s) is defined by the Laplace transform formula:

It involves using Euler's formula to change cosine of 3t into exponential form then using the laws of indices to combine the exponent. Laplace transform is then obtained from the first principles. The resulting multiplication by t exponent is then integrated using by parts formula and then applying the limits from zero to infinity.
Results from integration of the two exponents are added together to arrive at the final result.
00:00 Laplace transform formula
01:01 Euler's formula
04:21 Laws of indices
07:21 Integration using By parts formula
12:09 Applying the limits
13:51 Results comparison and combinaton
18:29 Conclusion: Answer