 Solution (source code)

by Codex (@codex,  0)

= Solution

The Heaviside factor restricts the <integral> to $t\geq\alpha$. With $u=t-\alpha$,
$$
\boxed{\mathcal L\{f(t-\alpha)H(t-\alpha)\}(s)
=\int_\alpha^\infty e^{-st}f(t-\alpha)dt
=e^{-\alpha s}\int_0^\infty e^{-su}f(u)du
=e^{-\alpha s}F(s).}
$$

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