#int_0^oo x^1000e^-x dx#
you can do this very quickly by noting that it is the equivalent of
#mathcal (L) { x^1000}_(s = 1}#
and as #mathcal(L) {t ^n} = (n!)/(s^(n+1))#
#int_0^oo x^1000e^-x = (1000!)/(1)^1001 = 1000!#
Or in terms of the gamma function:
#Gamma(n+1) = int_0^oo x^(n) e^(-x) dx = n!#
To generate your own solution you could start as per the factorial function with this
#color(blue)(int_0^oo e^(- alpha x) dx)#
#= [- 1/ alpha e^(- alpha x)]_0^oo color(blue)(= 1/alpha)#
#d/(d alpha) int_0^oo e^(- alpha x) dx = d/(dalpha) (1/ alpha)#
#implies int_0^oo (-x) e^(- alpha x) dx = - 1/ alpha^2# or #color(blue)( int_0^oo x e^(- alpha x) dx = 1/ alpha^2)#
Again # d/(d alpha) int_0^oo x e^(- alpha x) dx = d/(d alpha)( 1/ alpha^2)#
#implies int_0^oo (-x) x e^(- alpha x) dx = - 2/ alpha^3# or # color(blue)(int_0^oo x^2 e^(- alpha x) dx = 2/ alpha^3)#
Such that
# int_0^oo x^n e^(- alpha x) dx = (n!)/ alpha^(n+1)#