Laplace transform of t^n

I know the final result, its on all the charts. But I need to show step by step how to get the solution.

If someone could help out by getting me in the right direction or just plain giving me the answer, that would be much appreciated.

nicksauce
Homework Helper
Well probably a best way would be a proof by induction. Do you know what that is?

Edit: I'll add to this. You want to show $$\int_{0}^{\infty}t^ne^{-st}dt=\frac{n!}{s^{n+1}}$$ for all non-negative integers n. To do this, first show it is true for n=0. Then assume it is true for arbitrary n, and show that it is true for n+1, ie: $$\int_{0}^{\infty}t^ne^{-st}dt=\frac{n!}{s^{n+1}}\rightarrow\int_{0}^{\infty}t^{n+1}e^{-st}dt=\frac{(n+1)!}{s^{n+2}}$$. This can easily be done with an integration by parts.

Last edited:
HallsofIvy