How Can You Prove the Laplace Transform of t^n?

[seang]

Our professor asked us to derive an expression for the laplace transfrom of t^n. I did a few examples in MatLab and gathered that the Laplace Transform of t^n = n!/s^(n+1). I'm pretty sure this is correct, but I don't think my professor will be happy with it. I don't really know how I should go about proving it in a more sturdy way. I know I can integrate by parts for specific examples, but I'm not versed in integrating by parts with n's.

Any Suggestions?

 

[FredGarvin]

I don't see how you can get away without doing integration by parts. What is the definition of the Laplace transform?

$$L \{f(t) \} = \int_{0}^{\infty} f(t) e^{-st} dt$$
BTW, you know that the n's are constants, right?

 

[Corneo]

Let's say if you have

$$x(t) \iff X(s)$$

and you wish to find the Laplace transform of

$$t x(t)$$

Differentiate with respect to s of the Laplace transform integral. That is

$$\frac {d}{ds} \int_{0^-}^\infty t x(t) e^{-st} dt$$

You may move the derivative inside the integral and differentiate the exponential of the integrand.

Doing so you will see that $$t x(t) \iff - \frac {dX(s)}{s}$$

Try generalizing this for $t^n$. Note that for your specific problem

$$x(t) = t$$