Integrate[(1 + x/n)^n*Exp[-x], {x, 0, Infinity}]

Question

I evaluated

Integrate[(1 + x/n)^n*Exp[-x], {x, 0, Infinity}] . 

thinking the answer should be approximately $\sqrt{\pi n/2}$. Mathematica gave me

ConditionalExpression[E^n n ExpIntegralE[-n, n], Re[n] > 0 || n ∉ Reals]

which I am not sure how to interpret. How can I get Mathematica to give me something closer to what I was expecting?

Answer 1

Try

Assuming[Element[n, Integers] && n > 0,
 FunctionExpand@Integrate[(1 + x/n)^n Exp[-x], {x, 0, Infinity}]]

(* E^n n^-n Gamma[1 + n, n] *）