Because the function is not defined on [0,1]
That means we need to appeal to improper integrals.
You can write out e^x as 4 terms:
e^x = 1+x+x^2/2 + x^3/6
e^x - 1 = x+x^2/2 + x^3/6
(e^x-1)/x = 1 + x/2 + x^2 / 6
Now this is a polynomial you can integrate this easily.