I am new to Laplace transform, and have some hard time understanding the very first step of the "preparation" before taking the laplace transform.
$${f(t) =\int_0^t u \cosh(3u)\,\mathrm{d}u } =\frac13 \int_0^t u \,\mathrm{d} \sinh(3u) = 1/3 \bigg([u\sinh(3u)]^t_0 - 1/3[\cosh(3u)]^t_0\bigg) $$
How/why do we suddenly derivate cosine hyperbolic to sine hyperbolic out of nowhere? the two first lines are not clear.
On the third line we can see an integration by parts is done. But here again the integration of the cosine hyperbolic is done not the sine hyperbolic. So what was the goal of the first two steps?
All the preparative steps:
Hopefully my question is clear. I haven't found out yet how to put the zero in subscript on the third line.