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I can use the table directly, but I am struggling how to make the combination. This equation will help me to work on my gyroscope sensor equations ( magnetic based ) , My pain point is at the combination.

$$ t^n\mathrm{e}^{-at}\sin\omega t $$

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  • \$\begingroup\$ I’m voting to close this question because it's a homework question with no attempt shown. \$\endgroup\$ – Hearth Sep 2 '20 at 18:28
  • \$\begingroup\$ This is a duplicate of your previous question? What is the Laplace transform for this equation?. Edit and add details there. \$\endgroup\$ – brhans Sep 2 '20 at 19:13
  • \$\begingroup\$ A magnetic based gyroscope sensor? Can you provide a part # for that? \$\endgroup\$ – Voltage Spike Sep 3 '20 at 16:50
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  1. Find Laplace transform of sinωt i.e X(s)

2 .replace s by (s+a) in Laplace transform of sinωt ,you got X(s+a)

3.differentiate X(s+a) by n times with respect to s and that's you answer

Method for n times differentiation- 1.partial fraction X(s+a) ,here poles will be of complex form 2.and then differentiate both partial fractions by n times and after that combine then , hopefully you'll get your answer

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  • \$\begingroup\$ I'll add how you should differentiate n times ,I think you make some mistake there \$\endgroup\$ – user215805 Sep 2 '20 at 19:22
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The results using Mathematica.

LaplaceTransform[t^n Exp[-a t] Sin[w t], t, s]

$$\Gamma (n+1) \left((a+s)^2+w^2\right)^{\frac{1}{2} (-n-1)} \sin \left((n+1) \tan ^{-1}\left(\frac{w}{a+s}\right)\right)$$

It simplifies to rational functions with numeric values

With[{n = 2, a = 1, w = 3}, LaplaceTransform[t^n Exp[-a t] Sin[w t], t, s]]

$$\frac{18 \left(s^2+2 s-2\right)}{\left(s^2+2 s+10\right)^3}$$

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I used wxmaxima to calculate the problem, and that is the result:

enter image description here

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