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Integration by Partial fractions question

Mathematics
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\[\int\limits\limits\limits\limits_{}^{}\frac{ x ^{2}+x+2 }{ x ^{2}-1 } dx \]
Whenever I solve this I always get: A= -1 B= 2 then \[\int\limits\limits_{}^{} -\frac{ 1 }{ x+1 } + \frac{ 2 }{ x-1 } dx\] after substituting A and B and checking if I will still get the original integral, it's not the same anymore. How can I solve this integral?
did you do the long division first before u partial frac'ed it? u'll lose a few terms if you didn't do that.

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Other answers:

No. I immediately used partial fraction. Then I need to divide this first?
yes. you'll miss a few terms if u didn't(faced this problem once in my exam==costed me 5 precious minutes)
Oh. Ok, I'll try dividing it first.
This is a bit embarrassing, but I'm not sure if I got the quotient right. When I divide the two I get the answer of 1 with a remainder of \[\frac{ x+3}{ x ^{2}-1 }\]
yeah. then proceed to partial frac it. then, after that you can integrate it.
Just to be clear, after dividing I will get this? \[\int\limits_{}^{} \frac{ x+3 }{ x ^{2}-1 }\] I will then proceed to partial fraction then integrate? Thanks!
\(\int (1+\frac{x+3}{x^2 -1})dx \) .<--this yup. You're welcome :)
Oh, I ignored the 1. Thank you very much again! :D

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