# Determine the Laplace transform of f{{left({t}right)}}={t}^{4}-{t}^{2}{e}^{{{3}{t}}}+{2}

Determine the Laplace transform of
$f\left(t\right)={t}^{4}-{t}^{2}{e}^{3t}+2$
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Step 1
The given function is $f\left(t\right)={t}^{4}-{t}^{2}{e}^{3t}+2$
Find the Laplace transform of $f\left(t\right)={t}^{4}-{t}^{2}{e}^{3t}+2$ as shown below.
Step 2
It is known that,
$L\left\{1\right\}=\frac{1}{s}$
$L\left\{{t}^{n}\right\}=\frac{n!}{{s}^{n+1}}$
($L\left\{t\ne \left(at\right)\right\}=\frac{n!}{{\left(s-a\right)}^{n+1}}$
Then,
$L\left\{f\left(t\right)\right\}=L\left\{{t}^{4}-{t}^{2}{e}^{3t}+2\right\}$
$=L\left\{{t}^{4}\right\}-L\left\{{t}^{2}{e}^{3t}\right\}+L\left\{2\right\}$$=L\left\{{t}^{4}\right\}-L\left\{{t}^{2}{e}^{3t}\right\}+2L\left\{1\right\}$
$=\frac{24}{{s}^{5}}-\frac{2}{{\left(s-3\right)}^{3}}+\frac{2}{s}$

Step 3
Therefore, the Laplace transform of the function