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Is there someone who's taken complex analysis (or anyone that can see a stupid mistake in my math) that can explain why these results are different? If you plug them into the calculator, they come up differently, separated by exactly Pi/2. I've been straining my head on it but I can think of why. It may have something to do with the failure of log properties, or the fact that the answer to this integral is normally just arccot(x) + C, or that C isn't necessarily an arbitrary constant, but I don't know for sure.

The error was in my partial fraction expansion so now I get:



Which when I plug into my calculator does equal the 1/(-x^2-1). But when I do then integration after the fix, the result is still separated from the other answer by Pi/2

Well, I plugged in values beyond zero, and I moved the 1/2 into the log expression as a radical (so as not to need the absolute values). It seems to be the constant of integration that is the meddling factor. Your point on the nature of logarithmic integrals makes that clear. It may be that the constant isn't arbitrary.

It did; because comparing one integral to another is where I started. At first, I was just dealing with dx/(x^2+1), without the negatives to add the confusion. I simply set dx/(x^2+1)=dx/(x^2+1) and then had arctan on one side and the partial fraction expansion on the other. When I solved for C, and used an identity, it gave me arccot in terms of logarithme imaginaire. With the -dx/(x^2+1) calculation I was simply trying to do the same thing, expecting to get an answer that gave me arctan in the same terms.