Inverse Laplace Transform

Tips

Note if there is a $$(1-e^{-sT})$$ term in the denominator; if there is, put it to the side and note at the very end that the signal is right-side periodic with period $$T$$.
Pull out any exponentials in the numerator and put them to the side - these will cause time shifts in the time domain, which can be applied after taking the unshifted inverse.
If the remaining part of the transform is not a proper rational function (i.e. if the order of the numerator is not lower than the order of the denominator), use polynomial division or some other method to rewrite the transform as a collection of non-negative powers of $$s$$ followed by a "remainder" which is a proper rational function (i.e. the numerator is of a lower order than the denominator).
- The non-negative powers of $$s$$ will become impulses or derivatives of impulses in the time domain since $ \begin{align*} X(s)&\leftrightarrow x(t)\\ sX(s)&\leftrightarrow \frac{dx(t)}{dt}\\ a &\leftrightarrow a\,\delta(t)\\ a\,s^n&\leftrightarrow a\,\frac{d^n\delta(t)}{dt^n}\end{align*} $
- The proper rational part likely will likely need to be broken up using partial fraction expansion; the coefficients may be found in a variety of ways, including Heaviside's cover-up method.
Once you have the inverse of the parts of the transform not involving time shifts or periodicity, apply the time shifts as needed as well as the periodicity piece if needed.

Find the inverse Laplace transform of $$G_1(s)=\frac{s+20}{s^2+10s+21}$$.