anonymous
  • anonymous
Which method can help me to do the integral of (x^2+1)/(1-x^2)?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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AravindG
  • AravindG
quotient rule may be
nenadmatematika
  • nenadmatematika
write x^2+1 as x^2-1+2, make two basic integrals and solve
anonymous
  • anonymous
Because I just learned in school to do a partial fraction decomposition but this somehow does not work out very well. I´ll use your tip nenadmatematika.

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dumbcow
  • dumbcow
ha i was just going to suggest partial fractions 1-x^2 = (1-x)(1+x)
dumbcow
  • dumbcow
oh right split it into 2 fractions first
nenadmatematika
  • nenadmatematika
\[=\int\limits_{?}^{?}(x^2-1)/(1-x^2)dx+\int\limits_{?}^{?}2/(1-x^2)=-\int\limits_{?}^{?}dx+2\int\limits_{}^{?}dx/(1-x^2)\]\[=-x+\ln \left| (1+x)/(1-x) \right|+c\]
anonymous
  • anonymous
This kind of idea comes from experience I guess?
nenadmatematika
  • nenadmatematika
yes, it takes some practice to get the routine, but consider it very easy integral...:D
anonymous
  • anonymous
ok, thx for your help :)
nenadmatematika
  • nenadmatematika
if you've got some more problems with integrals hit me now :D
TuringTest
  • TuringTest
Do you want to see it with partial fractions? it's not very hard
anonymous
  • anonymous
Yes, might be a nice to see a different approach.
TuringTest
  • TuringTest
well actually, I guess nenad did use them it do the second integral (or at least I don't see how he did without them) another way to get to that point it to rearrange the fraction so you can perform long division\[\frac{1+x^2}{1-x^2}=\frac{-x^2-1}{x^2-1}=-1-\frac2{x^2-1}\]now doing partial fractions on the second term\[\frac{-2}{x^2-1}=\frac A{x-1}+\frac B{x+1}\]\[A(x+1)+B(x-1)=-2\]\[A=-1,B=1\]so we wind up with\[\int-1+\frac1{x-1}-\frac1{x+1}dx\]again
nenadmatematika
  • nenadmatematika
well, that's nice but you would you make that second part complicated when you know that it's a basic integral: \[\int\limits_{?}^{?}dx/(1-x^2)=(1/2)\ln \left| (1+x)/(1-x)\right|\]
TuringTest
  • TuringTest
I didn't know that form, but if they are supposed to use partial fractions in the problem they must do it here.
nenadmatematika
  • nenadmatematika
OK, I didn't know that they must do it like that :D
anonymous
  • anonymous
Can I do this partial fractions stuff also without changing x^2+1 to x^2-1+2?
TuringTest
  • TuringTest
the order of the numerator has to be smaller than that in the denominator to do partial fractions, so you need to make that happen somehow long division is a very standard procedure in these cases, but you can also do what nenad did frequently.
TuringTest
  • TuringTest
the degree of the numerator*
anonymous
  • anonymous
Ah, Thank you. We only talked in school about the cases where the degree is higher or lower but never about what it means when they have the same degree. :( My teacher sucks... Thank you TuringTest for your help and explanation.

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