Where the initial variable is $t$ and the transform variable is $s$. And $w$ is a constant.
I'm trying to solve this problem but can't seem to make any progress. I've tried from the definition, but that didn't get me very far. Would really appreciate some help with this.
Result from WolframAlpha
Hint:
By simply writing out the definition: $$F(s)=\int_0^\infty t^n\cosh(wt)e^{-st}dt=\int_0^\infty t^n\left(\frac{e^{wt}+e^{-wt}}{2}\right)e^{-st}dt \\=\frac{1}{2}\left(\int_0^\infty t^n{e^{(w-s)t}+\int_0^\infty t^ne^{-(w+s)t}}dt\right)$$ Now recall the integral representation of the Gamma function: $$\Gamma(n)=\int_0^\infty t^{n-1}e^{-t}dt.$$ Now do an obvious substitution and you are good to go.