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.