Question

the gamma function r(x) is defined by r(x)= integral from 0 to infinity of x^n-1(e^-1)dx. find r(1), r(2), r(3)

Answer 1

actually,

Γ(n) = ∫[0,∞] x^(n-1) e^(-x) dx
which is unusual, since n is usually used for constants. Usually it is written
Γ(t) = ∫[0,∞] x^(t-1) e^(-x) dx

As you probably should know, Γ(n) = (n-1)! for integer n.

So,
Γ(1) = 1
Γ(2) = 1
Γ(3) = 2