# Find the Laplace transformation (evaluating the improper integral that defines this transformation) of the real valued function f(t) of the real varia

Find the Laplace transformation (evaluating the improper integral that defines this transformation) of the real valued function f(t) of the real variable t>0. (Assume the parameter s appearing in the Laplace transformation, as a real variable).
$f\left(t\right)=2{t}^{2}-4\mathrm{cosh}\left(3t\right)+{e}^{{t}^{2}}$
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Step 1

here $f\left(t\right)=2{t}^{2}-4\mathrm{cosh}\left(3t\right)+{e}^{{t}^{2}}$
we know that
$L\left\{{t}^{n}\right\}=\frac{n!}{{s}^{n+1}},L\left\{\mathrm{cosh}\left(at\right)\right\}=\frac{s}{{s}^{2}-{a}^{2}}$
$L\left\{{e}^{{t}^{2}}\right\}=F\left(\frac{s}{2}\right)-\frac{1}{2}i\sqrt{\pi }\cdot {e}^{-\frac{{s}^{2}}{4}}$
Also , $L\left\{f\left(t\right)+g\left(t\right)+h\left(t\right)\right\}=L\left\{f\left(t\right)\right\}+L\left\{g\left(t\right)\right\}+L\left\{h\left(t\right)\right\}$
Taking Laplace transform of eq
$L\left\{f\left(t\right)\right\}=L\left\{2{t}^{2}-4\mathrm{cosh}\left(3t\right)+{e}^{{t}^{2}}\right\}$
$=2L\left\{{t}^{2}\right\}-4L\left\{\mathrm{cosh}\left(3t\right)\right\}+L\left\{{e}^{{t}^{2}}\right\}$
$=2\cdot \frac{2!}{{s}^{2+1}}-4\frac{s}{{s}^{2}-{3}^{2}}+F\left(\frac{s}{2}\right)-\frac{1}{2}i\sqrt{\pi }\cdot {e}^{-\frac{{s}^{2}}{4}}$
$L\left\{f\left(t\right)\right\}=\frac{4}{{s}^{3}}-\frac{4s}{{s}^{2}-9}+F\left(\frac{s}{2}\right)-\frac{1}{2}i\sqrt{\pi }\cdot {e}^{-\frac{{s}^{2}}{4}}$
Step 2
This is required Laplace transformation of given f(t).