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

Working

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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Categories: Laplace Transforms Workings

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