- #1

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## Homework Statement

Using:

[tex]

\mathcal{L}\big\{t^n\big\}=\frac{n!}{s^{n+1}}\text{for all s>0}[/tex]

Give a formula for the Laplace transform of an arbitrary nth degree polynomial

[tex]

p(t)=a_0+a_1t^1+a_2t^2+...+a_nt^n[/tex]

## Homework Equations

[tex]\mathcal{L}=\lim_{b\rightarrow\infty}\int_{0}^{b}p(t)e^{-st}dt[/tex]

## The Attempt at a Solution

I figured that the function p(t) is a sum and can be expressed as:

[tex]\sum_{i=0}^{n}a_it^i[/tex]

And integration gives:

[tex] \lim_{b\rightarrow\infty}\int_{0}^{b}\Bigg(\sum_{i=0}^{n}a_it^i\Bigg)e^{-st}dt[/tex]

I'm not quite sure what to do next, I don't recall if I can move the integral and limit inside of the summation or not without being arrested by the math police for violating math laws.