# Looking for a proof

Hey guys, how can this Laplace transform be proven. I always see it in the tables but don't know how it came to be.

$$L[t^n] = \frac{n!}{s^{n+1}}$$

Just do the transformation. With a simple change of variables you can reduce the integral to the gamma function.

Ah. Ok I got it thanks. I guess I must have let it slip by me that gamma function is related.

here is a formal proof:
You can show that
$$\int t^n e^{-ts} dt = -s^{-n-1}\int_{st}^{\infty}x^n e^{-x}dx +c$$
therefore
$$\int_{0}^{\infty} t^n e^{-ts} dt =s^{-n-1}\int_{0}^{\infty}x^n e^{-x}dx =s^{-n-1} n!$$