real analysis - Euler and the factorial function

I recently purchased H. M. Edwards' book entitled The Riemann Zeta Function. In the early pages of the volume, concerning the factorial function $\Gamma$, Edwards notes that

"Euler observed that $\Gamma(n)=\int_0^\infty e^{-x}x^{n-1}dx$."

My question is twofold:

1. How does one "observe" such a formula? Surely, this does not merely come from an intuitive observation?


2. How does one prove this formula, and more importantly, where does the techincal motivation for it come from?

Answer

If you have not seen integration by parts before, it is strongly related to the product rule of differentiation.

$$\frac{d}{dx}x^ne^{-x}=nx^{n-1}e^{-x}-x^ne^{-x}\\ \left.x^ne^{-x}\right|_0^{\infty}=\int_0^{\infty}nx^{n-1}e^{-x}dx-\int_0^{\infty}x^ne^{-x}dx\\ \int_0^{\infty}x^ne^{-x}dx=n\int_0^{\infty}x^{n-1}e^{-x}dx$$
So the integral with $n$ is related to the integral with $n-1$; and by the same rule that $n!$ is.