I'm learning Laplace transforms and because my textbook is rubbish I've gone to external sources, namely this website: Paul's Online Math Notes. It's good for the most part, but there is an example that I really don't understand. Here it is:
And here are the transforms they name:
What I don't understand is how they used #32 and #5. The integral step is very confusing for me. Can someone help me to understand what it is they did?
Recognize that $t^{3/2}$ may be cleverly expressed as an integral of $\sqrt{t}$ and you can use the integral formula in rule 32 in conjunction with rule 5.
The integral formula says that the transform of an integral on $[0, t]$ is the transform of the integrand divided by $s$.