Integration and natural log question WolframAlpha is confusing me. I'm working on two different integral problems, and with each one, for some reason, Wolfram switches the signs of the answer.


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*http://www.wolframalpha.com/input/?i=integral+dx/(x^.5(x-1))

*http://www.wolframalpha.com/input/?i=integral+(x-1)/(x^2-4)


For the first one, they get $\log[1 - \sqrt{x}] - \log[(1 + \sqrt{x})]$ (see under alternate forms, thats the one I'm using).
However, I get:
$\log[\sqrt{x}-1] - \log[(\sqrt{x}+1)]$ And so it differs by a sign, but its really bothering me, since I'm sure I'm doing it right. And the same applies to the second one:
They get:
$3/4 \log(-x-2)+1/4 \log(x-2)$
I get:
$3/4 \log(x+2)+1/4 \log(x-2)$
And so it differs by a sign again. (If this is a common mistake, then maybe you can point it out, but if this is just me and you can't see where I could have gone wrong, I can list my steps)

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*The (real) indefinite integral is actually different in $(0, 1)$ and in $(1, \infty)$.  WolframAlpha gave you the answer in the first interval and your answer works for the second.  (Remember that in the real numbers, the logarithm of a non-positive number is undefined.)

*This time WolframAlpha's answer is not defined for any real $x$.  Your answer works in $(2, \infty)$ but there is a different antiderivative in $(-2, 2)$ and yet another antiderivative in $(-\infty, -2)$.
This is a fairly subtle property of the logarithm.  Remember that any two antiderivatives of a function differ by a constant.  The problem in this case is that the constant is complex, so if you aren't working in the complex numbers you get genuinely different answers.
