Solve Laplace Transform: L{tcos(t)}

[Gogeta007]

Homework Statement

L{tcos(t)}

Homework Equations

Using the laplace transform find the equation.

The Attempt at a Solution

I already have a really long answer. . .I was just wondering if someone can explain this:

Another method (appart from using integration by parts multiple times) was mentioned by my professor, he said to take the partial with respect to s of the integral, which will then just leave the partial laplace transform of cos(t) which can be found trough tables

I remember he explained it but now that I think about it it doesn't make sense:

L {t cos(t)} = (int)^infinity₀ e^-stt cos(t)dt
then you can take the partial with respect to s (d/ds) of that integral, and yo uwould end up with

d/ds [L{cost}] = d/ds [s/(s^2+1)]


cans omeone explain?
I remember he mentioned the potential function

 

[vela]

You use the fact that

$$te^{-st} = -\frac{\partial}{\partial s}e^{-st}$$

and then switch the order of integration and differentiation.