Respuesta :

By laplace transform definition of { t^n * f(t) } 

(-1)^n * ( d^n/ds^n ) F(s) 

t^2*sin(2t) <==== let's apply it 

(-1)^2 * ( d^2/ds^2 ) ( 2 / (s^2 + 2^2) ) 

1 * ( d^2/ds^2 ) ( 2 / (s^2 + 4) ) <===== let's find the first and second derivative : 

( (s^2 + 2^2) * 0 - 2 * (2s) / (s^2 + 4)^2 ) 

( - 4s) / (s^2 + 4)^2 ) <==== let's find the second derivative 

( (s^2 + 4)^2 * -4 - - 4s * 2 * (s^2 + 4) * 2s ) / (s^2 + 4)^4 ) 

( -4(s^2 + 4)^2 + 16s^2 * (s^2 + 4) ) / (s^2 + 4)^4 ) 

( -4(s^2 + 4) + 16s^2 ) / (s^2 + 4)^3 ) 

( -4s^2 - 16 + 16s^2 ) / (s^2 + 4)^3 ) 

( 12s^2 - 16 ) / (s^2 + 4)^3 )

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Laplace transformation is the transformation in the integral form which convert a function of a real variable to a complex variable.

The Laplace transform of the given function is,

[tex]L[t^{2}\sin(2t)]=\left [\dfrac{4(3s^2-4)}{(s^2+4)^2} \right ]\\[/tex]

What is Laplace transformation?

Laplace transformation is the transformation in the integral form which convert a function of a real variable to a complex variable.

Given information-

The function given in the problem is,

[tex]t^{2}\sin(2t)[/tex]

By the Laplace transform we can write that,

[tex]L[t^{2}\sin(2t)]=(-1)^2\dfrac{d^2}{ds^2}L(\sin 2t)\\L[t^{2}\sin(2t)]=\dfrac{d}{ds}\;\dfrac{d}{ds}\dfrac{2}{s^2+4}\\L[t^{2}\sin(2t)]=\dfrac{d}{ds}\left [\dfrac{-4s}{(s^2+4)^2} \right ]\\[/tex]

Solve further to get the final result,

[tex]L[t^{2}\sin(2t)]=\left [\dfrac{(s^2+4)^2(-4)+4s(s^2+4)\times2s}{(s^2+4)^2} \right ]\\[/tex]

Simplify the above equation as,

[tex]L[t^{2}\sin(2t)]=\left [\dfrac{4(3s^2-4)}{(s^2+4)^2} \right ]\\[/tex]

Hence, the Laplace transform of the given function is,

[tex]L[t^{2}\sin(2t)]=\left [\dfrac{4(3s^2-4)}{(s^2+4)^2} \right ]\\[/tex]

Learn more about the Laplace transform here; https://brainly.com/question/17062586

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