Could someone show me explicitly what is going in the first line? I am confused as I have learnt Laplace transformations but I cannot figure out which Laplace transformation this is.. and the (a) in the first line, is that variable of the Laplace?
2026-03-29 07:39:21.1774769961
Anyone has any idea on what is going on in this method of integration?
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The Laplace transform of a function $f(t)$ is defined as
$$\mathcal{L}\{f\}(a)=\int_0^\infty f(t)e^{-at}\,dt$$
where $a$ denotes the parameter of the integral (i.e. $a$ appears in the evaluation).
In your case, for the first line, they set $f(t)=\sin(bt)/t$.
For the second equality, the frequency domain integration property was used.