# calculate the Inverse Laplace Transform of of the following function: Use partial fractions where needed:F(s)= 1/s^5

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lfryerda | High School Teacher | (Level 2) Educator

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The definition of the Laplace transform is L[f](s)=\int_0^infty e^{-st}f(t)dt.

This means that for f(t)=1 its Laplace transform is L[1](s)=int_0^infty e^{-st}dt

=lim_{k->infty} int_0^k e^{-st}dt

=lim_{k->infty}1/s(1-e^{-sk})

=1/s

If we let f(t)=t then its Laplace transform is

L[t](s)=lim_{k->infty}int_0^k te^{-st}dt

=lim_{k->infty}(1/s^2e^{-ks}(-sk-1)+1/s^2)   using integration by parts

=1/s^2

Continuing in the same way using integration by parts and simplifying each expression, we see that L[t^4](s)={4!}/{s^5}

This means that the Inverse Laplace Transform of F(s)=1/s^5 is t^4/{4!}.

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