# 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 | Certified Educator

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!}`.**