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Find the Laplace of {$e^{rt}$}

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Definition of Laplace Transform:

{$$ F(s)=\int _0^\infty f(t)e^{-st}\, dt\ $$}

To find the Laplace transform of {$ f(t) = e^{rt} $}

{$$ \begin{align} F(s)=\int _0^\infty e^{rt}e^{-st}\, dt\ \cr \end{align} $$}

Integration by parts:

{$$ \large \int uv' dx = uv - \int u'v dx \; $$}

  '
{$ u $}{$\displaystyle \quad e^{rt} \quad$}{$\displaystyle \quad r\, {{e}^{r t}}\quad$}
{$ v $}{$\displaystyle \quad -\frac{{{e}^{-s t}}}{s}\quad $}{$\displaystyle \quad e^{-st}\quad $}

{$$ \large \int _0^\infty uv' dt = \left.uv\right| _0^\infty - \int _0^\infty u'v dt \; $$}

{$$ \large \int _0^\infty e^{rt}e^{-st} dt = \left.e^{rt}\left(-\frac{{{e}^{-s t}}}{s}\right)\right| _0^\infty - \int _0^\infty r\, {{e}^{r t}}\left(-\frac{{{e}^{-s t}}}{s}\right) dt \; $$}


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