Question

determine a transformada de Laplace FT=3et+cos3t​

Answer 1

A transformada de Laplace da função dada é igual a:

    $\Large\displaystyle\begin{gathered}\mathcal{L}\left\{3et+\cos3t\right\}=\frac{3e}{s^2}+\frac{s}{s^2+9}\end{gathered}$

A transformada de Laplace é dada da seguinte forma:

    $\Large\displaystyle\text{ \begin{gathered} \underline{\boxed{F(s)=\mathcal{L}\{f(t)\}=\int _0^{\infty} e^{-st}\cdot f(t)dt}}\end{gathered} }$

Com isso, podemos calcular a transformada de et e a do cos wt, mas as mesma já são tabeladas, logo:

  $\Large\displaystyle\text{ \begin{gathered}\underline{\boxed{\mathcal{L}\{et\}=\frac{e}{s^2} }}\ \ \wedge \ \ \underline{\boxed{\mathcal{L}\{\cos \omega t\}=\frac{s}{s^2+\omega^2}}} \end{gathered} }$

Substituindo, temos:

$\Large\displaystyle\begin{gathered}\mathcal{L}\left\{3et+\cos3t\right\}= 3\cdot\mathcal{L}\{et\}+\mathcal{L}\{\cos3t\}\end{gathered}$

$\Large\displaystyle\begin{gathered}\mathcal{L}\left\{3et+\cos3t\right\}= 3\cdot \frac{e}{s^2} +\frac{s}{s^2+3^2} \end{gathered}$]

$\Large\displaystyle\text{ \begin{gathered}\therefore \green{\underline{\boxed{\mathcal{L}\left\{3et+\cos3t\right\}= \frac{3e}{s^2} +\frac{s}{s^2+9} }}}\ \ (\checkmark).\end{gathered} }$

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