# determine the inverse Laplace transform of the function. L^{-1}\{R(s)\}=L^{-1}\{\frac{7}{(s+3)(s-3)}\}

determine the inverse Laplace transform of the function.
${L}^{-1}\left\{R\left(s\right)\right\}={L}^{-1}\left\{\frac{7}{\left(s+3\right)\left(s-3\right)}\right\}$

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Step 1
Given function is,
$R\left(s\right)=\frac{7}{\left(s+3\right)\left(s-3\right)}$
This can be re-written as,
$R\left(s\right)=\frac{7}{6}\left[\frac{1}{s-3}-\frac{1}{s+3}\right]$
Step 2
Inverse Laplace is given by,

Step 3
Hence, Laplace inverse of the given function is,
${L}^{-1}\left\{R\left(s\right)\right\}=\frac{7}{6}\left[{L}^{-1}\left\{\left[\frac{1}{s-3}\right]\right\}-{L}^{-1}\left\{\frac{1}{s+3}\right\}\right]$
$=\frac{7}{6}\left({e}^{3t}-{e}^{-3t}\right)$