# Find the inverse Laplace transform of: (2s)/((s+1)^2+4)

Find the inverse Laplace transform of: $\frac{2s}{\left(s+1{\right)}^{2}+4}$
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allelog5
Hint:
Note that
${\mathcal{L}}^{-1}\left(\frac{2s}{\left(s+1{\right)}^{2}+4}\right)=2{\mathcal{L}}^{-1}\left(\frac{s+1}{\left(s+1{\right)}^{2}+4}-\frac{1}{\left(s+1{\right)}^{2}+4}\right).$
Using linearity, it is thus enough to find the inverse Laplace Transform of $\frac{s+1}{\left(s+1{\right)}^{2}+4}$ and of $\frac{1}{\left(s+1{\right)}^{2}+4}$. To find these, use standard inverse Laplace transforms along with the fact that
${\mathcal{L}}^{-1}\left(F\left(s-a\right)\right)={e}^{at}f\left(t\right),$
where F is the Laplace transform of f (with $a=-1$ in your problem).