Laplace Transform to evaluate $\int_{0}^{\infty} e^{-2t} *t*\sin(4t)dt$

It's rigorous (IMO), and also very clever!

See, instead of using integration by parts, if you know about the Lapalce transform, you can use it! Just because a person's solution spans multiple pages does not mean another approach is "wrong." I always tell my students that sometimes going the long way means you're doing it right (in some contexts).

In general, if $f(t)$ admits a Laplace transform, then you can avoid integration by parts in integrating $f$ against $e^{-st}$ via the Laplace transform.