# What is the laplace transform of the following function f(t)=t^(10) cosh (t)

What is the laplace transform of the following function
$f\left(t\right)={t}^{10}\mathrm{cosh}\left(t\right)$
Is there another method that doesn't include differentiating the Laplace transform of $\mathrm{cosh}t$ ten times?
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Hajdiniax
$\begin{array}{rl}f\left(t\right)& ={t}^{10}\mathrm{cosh}t\\ & ={t}^{10}\left(\frac{{e}^{t}+{e}^{-t}}{2}\right)\\ & =\frac{1}{2}{t}^{10}{e}^{t}+\frac{1}{2}{t}^{10}{e}^{-t}\\ \mathcal{L}\left(f\left(t\right)\right)& =\frac{1}{2}\cdot \frac{10!}{\left(s-1{\right)}^{11}}+\frac{1}{2}\cdot \frac{10!}{\left(s+1{\right)}^{11}}\end{array}$

tophergopher3wo
Yes. Remember that $\mathrm{cosh}\left(t\right)=\frac{{e}^{t}+{e}^{-t}}{2}$. Use this to rewrite f(t) and use a relatively simple Laplace transform for functions of the form ${t}^{n}{e}^{at}$