Find the inverse Laplace transforms of the following functions:

Note that as soon as we step inside the exam hall we write down the following rule that is on the tables but not nearly in as nice a form:





This looks quite tricky but all it needs is a little trick. Multiply out (s+1)^3 by hand (or by memory or by the Binomial Theorem (in tables)) to get:


Now taking advantage of the linearity of the inverse Laplace transform (I’m just going to write K instead of \mathcal{L}^{-1}. To get a nice inverse Laplace transform symbol I have to type \mathcal{L}^{-1} — along with some dollar signs and I’m sick of it! — no keyboard shortcuts on this page — the notes have shortcuts.)

K\left\{\frac{1}{s}\right\}+3 K\left\{\frac{1}{s^2}\right\}+3K\left\{\frac{1}{s^3}\right\}+K\left\{\frac{1}{s^4}\right\}.

Now using the tables;





Now this looks like


which we know how to deal with. It is our job to get it looking a bit more like this. We can do this by taking out a 4 out of the bottom line:


Now using linearity, and the tables, this has inverse transform: