# Find the inverse laplace of this function: F(s)=(s+8)/(s^2+4s+5)

Find the inverse laplace of this function: $F\left(s\right)=\frac{s+8}{{s}^{2}+4s+5}$
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Kristin Myers
Hint. You may observe that
$\mathcal{L}\left({e}^{-at}\mathrm{cos}\omega t\right)=\frac{s+a}{\left(s+a{\right)}^{2}+{\omega }^{2}},$
$\mathcal{L}\left({e}^{-at}\mathrm{sin}\omega t\right)=\frac{\omega }{\left(s+a{\right)}^{2}+{\omega }^{2}}.$
Then apply it to
$F\left(s\right)=\frac{s+2}{\left(s+2{\right)}^{2}+{1}^{2}}+6\phantom{\rule{mediummathspace}{0ex}}×\frac{1}{\left(s+2{\right)}^{2}+{1}^{2}}.$
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Lara Cortez
Divide it as s+2 and 6.
$\mathcal{L}\left({e}^{-at}\mathrm{cos}\omega t\right)=\frac{s+a}{\left(s+a{\right)}^{2}+{\omega }^{2}}.$