# 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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- Wiki Commons Believed to be in the public domain.
- Fooplot manually enhanced

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** Categories:** Laplace Transforms Workings

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