Inverse Laplace Transforms

The inverse Laplace transform is simply the trip back: given $F(s)$, find the function $f(t)$ it came from. "Inverse" means it undoes the transform, the way subtracting undoes adding. It's what you need at the end of every problem, because when we solve equations with transforms the answer shows up as some $F(s)$. In practice you're just reading the table backwards. The formulas look busy, but they're just three pairs: $\frac{1}{s - a}$ (comes from $e^{at}$), and the two with $(s - a)^2 + b^2$ on the bottom (come from $e^{at}\cos bt$ and $e^{at}\sin bt$, depending on what's on top). There's no nice integral for going back by hand, and $F(s)$ rarely looks exactly like a table entry, so most of the work is ordinary algebra that rearranges it until it does. Bear with the algebra; the idea itself is simple.

Partial fractions split a rational function into simple pieces: $\dfrac{s + 7}{(s - 3)(s + 2)} = \dfrac{2}{s - 3} - \dfrac{1}{s + 2}$, which inverts to $2e^{3t} - e^{-2t}$. This is the same technique you used in calculus to integrate rational functions. Each factor $s - a$ in the denominator turns into an $e^{at}$, so the roots of the denominator tell you which exponentials appear.

Irreducible quadratics: complete the square to get $(s - a)^2 + b^2$, then split the numerator into a multiple of $(s - a)$ plus a constant. Those two pieces match the cosine and sine entries.

Adjust constants: $\frac{1}{s^2 + 4} = \frac12\cdot\frac{2}{s^2 + 4}$ inverts to $\frac12\sin 2t$. The table wants a $2$ on top, so you put one there and pay for it with a $\frac12$ out front.

A common mistake. It's tempting to invert $\frac{1}{s^2 + 4}$ as $\sin 2t$, because the denominator matches. It isn't quite: the table entry is $\frac{b}{s^2 + b^2}$, so $\sin 2t$ goes with $\frac{2}{s^2 + 4}$. Always check the numerator too, and fix the constant the way we did above.

After designing a system in the s-domain, engineers invert the transform to see the actual time response: overshoot, settling time, oscillation.