#### Question

Evaluate the given Laplace transform without evaluating the integral. $\mathscr{L}\left\{t^{2} * t e^{t}\right\}$

Verified

#### Step 1

1 of 2

Laplace transform of given function is

$\mathcal{L}\left\{t^{2}\star te^{t} \right\}$

According to convolution theorem $7.9$

$\mathcal{L}\left\{ f \star g\right\} =\mathcal{L}\left\{ f\right\}\mathcal{L}\left\{ g\right\}=F(s)G(s)$

Where $f$ and $g$ be piecewise continuous on $[0,\infty)$ and of exponential order. Thus

$\mathcal{L}\left\{t^{2} \star te^{t} \right\}=\mathcal{L}\left\{t^{2} \right\}\cdot \mathcal{L}\left\{te^{t} \right\}$

Since, we know $\mathcal{L}\left\{ t^{n}\right\}=\dfrac{n!}{s^{n+1}}$ and $\mathcal{L}\left\{ e^{at}f(t)\right\}=F(s)\bigg|_{s \rightarrow s-a}$

$=\dfrac{2!}{s^{3}}\cdot \dfrac{1}{s^{2}}\bigg|_{s \rightarrow s-1}$

$=\dfrac{2!}{s^{3}}\cdot \dfrac{1}{(s-1)^{2}}$

$=\dfrac{2}{s^{3}(s-1)^{2}}$

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