# Find the Laplace Transform of the function f(t) = e^{at}

Find the Laplace Transform of the function
$f\left(t\right)={e}^{at}$
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2abehn
Step 1
We have to find the Laplace transform of the function f(t) = e^(at)
we also know that
$F\left(s\right)=L\left\{f\left(t\right)\right\}$
Step 2
doing the transforms we get
$F\left(s\right)=L\left\{f\left(t\right)\right\}$
$=L\left\{{e}^{at}\right\}$
$={\int }_{0}^{\mathrm{\infty }}{e}^{-st}{e}^{at}dt$
$=\underset{r\to \mathrm{\infty }}{lim}{\int }_{0}^{r}{e}^{-st+at}dt$
$=\underset{r\to \mathrm{\infty }}{lim}{\int }_{0}^{r}{e}^{\left(a-s\right)t}dt$
$=\underset{r\to \mathrm{\infty }}{lim}|\frac{{e}^{\left(a-s\right)t}}{\left(a-s\right)}{|}_{0}^{r}$
$=\frac{{e}^{\left(a-s\right)\mathrm{\infty }}}{\left(a-s\right)}-\frac{{e}^{\left(a-s\right)0}}{\left(a-s\right)}$
$=-\frac{1}{\left(a-s\right)}$
$=\frac{1}{\left(s-a\right)}$
Step 3
Laplace Transform of the function $f\left(t\right)={e}^{at}$ is $\frac{1}{\left(s-a\right)}$