Indefinite Integral

You really don't need to solve this problem to solve your other problem in the other thread. But I'll try to find a technique that works just for the heck of it.

(Check my math for errors.)

Let A = ∫ √(1+1/x) dx. (This is what we want to integrate.)

Let r = √(1+1/x).

Solve for x:
r^2 = 1 + 1/x
r^2 - 1 = 1/x
x = 1/(r^2 - 1)

So dx = -1/(r^2 - 1)^2 * 2r dr.
And √(1+1/x) = r.

Thus A = ∫ [-2 r^2 / (r^2 - 1)^2] dr.

You can probably finish the job using the method of rational functions.