Your two equations should be
$\displaystyle A+B=0,$
$\displaystyle Ab+Ba=1.$
You got a minus sign in there that doesn't belong.
So, what is the solution to this system?
I'm not sure your integrations are correct. I think that, yet again, you're off by a minus sign. The rule goes like this:
$\displaystyle \displaystyle\int\frac{1}{x-a}\,dx=\ln|x-a|+C,$ but
$\displaystyle \displaystyle\int\frac{1}{a-x}\,dx=-\ln|x-a|+C.$
How does that change things?
No, no. The logarithms don't simplify that way. Exponentiation does NOT distribute over addition (most functions don't). I would use the logarithm identities to help here:
$\displaystyle \dfrac{-\ln(a-x)+\ln(b-x)}{b-a}=\dfrac{\ln((b-x)/(a-x))}{b-a}.$
Now exponentiate.