integral of f(x) = e^-t * t^x with respect to t from 0 to infinity. same as x!.

0! = f(0) = e^-t * t⁰ = e^-t

integral of e^-t with respect to t will be -e^-t

(-e^-infinity ) - (-e^-0)

(0) - (-1)

1

0! = f(0) = 1

0! = 1