The integral you gave is not a good integral.

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

Then,

e^x - 1 = x+x^2/2 + x^3/6

Thus,

(e^x-1)/x = 1 + x/2 + x^2 / 6

Now this is a polynomial you can integrate this easily.