TITLE: Prove the existence of Laplace transform of $2te^{t^2}\cos(e^{t^2})$
QUESTION [10 upvotes]: I tried to directly compute the integral, but I was unable to and wolfram alpha says it cannot find the answer in terms of elementary integrals. How can I prove the existence of the Laplace transform without directly computing it? Helpful hints leading me to the answer will be accepted as the answer(and I actually prefer this to outright stating it. Any help is appreciated!
I messed up and put the wrong function in the title initially- it should be the derivative of the function I gave. I am sorry for wasting people's time.

REPLY [3 votes]: Notice that 
$$2te^{t^2}\cos(e^{t^2}) = \dfrac{d}{dt} \sin(e^{t^2})$$
And that if 
$$\mathscr{L}\{f(t)\} = F(s)$$
then
$$\mathscr{L}\{f'(t)\} = sF(s) -f(0)$$
and also maybe apply @Mark_Viola's rationale for why F(s) must exist.