# 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.

1. http://www.wolframalpha.com/input/?i=integral+dx/(x^.5(x-1))
2. 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)

-

1. 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.)
2. 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 $(-\infty, -2)$.