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Integrals

Question
CBSEENMA12035668

Evaluate:
integral fraction numerator straight x squared over denominator straight x to the power of 4 plus straight x squared minus 2 end fraction dx

Solution
integral fraction numerator straight x squared over denominator straight x to the power of 4 plus straight x squared minus 2 end fraction dx
equals integral fraction numerator straight x squared over denominator left parenthesis straight x squared minus 1 right parenthesis thin space left parenthesis straight x squared plus 2 right parenthesis end fraction dx
equals integral fraction numerator straight x squared over denominator left parenthesis straight x minus 1 right parenthesis thin space left parenthesis straight x plus 1 right parenthesis thin space left parenthesis straight x squared plus 2 right parenthesis end fraction dx
Using space partial space fraction comma
space space fraction numerator straight x squared over denominator left parenthesis straight x minus 1 right parenthesis left parenthesis straight x plus 1 right parenthesis thin space left parenthesis straight x squared plus 2 right parenthesis end fraction space equals space fraction numerator straight A over denominator left parenthesis straight x minus 1 right parenthesis end fraction plus fraction numerator straight B over denominator left parenthesis straight x plus 1 right parenthesis end fraction plus fraction numerator Cx plus straight D over denominator left parenthesis straight x squared plus 2 right parenthesis end fraction
fraction numerator straight x squared over denominator left parenthesis straight x minus 1 right parenthesis left parenthesis straight x plus 1 right parenthesis left parenthesis straight x squared plus 2 right parenthesis end fraction equals fraction numerator straight A left parenthesis straight x plus 1 right parenthesis left parenthesis straight x squared plus 2 right parenthesis plus straight B left parenthesis straight x squared plus 2 right parenthesis left parenthesis straight x minus 1 right parenthesis plus left parenthesis Cx plus straight D right parenthesis left parenthesis straight x minus 1 right parenthesis left parenthesis straight x plus 1 right parenthesis over denominator left parenthesis straight x minus 1 right parenthesis left parenthesis straight x plus 1 right parenthesis left parenthesis straight x squared plus 2 right parenthesis end fraction
Equating the coefficients from both the numerators we get,
 A+B+C = 0 ....(1)
  A-B+D = 1 ....(2)
  2A+2B-C = 0 ...(3)
  2A-2B-D = 0 .....(4)
Solving the above equations we get,
 straight A equals 1 over 6 comma space space straight B space equals space minus 1 over 6 comma space straight C equals space space 0 comma space straight D space equals space 2 over 3
Our space Intergral space becomes comma
integral fraction numerator straight x squared over denominator left parenthesis straight x minus 1 right parenthesis thin space left parenthesis straight x plus 1 right parenthesis space left parenthesis straight x squared plus 2 right parenthesis end fraction dx space equals space integral fraction numerator 1 over denominator 6 left parenthesis straight x minus 1 right parenthesis end fraction minus fraction numerator 1 over denominator 6 left parenthesis straight x plus 1 right parenthesis end fraction plus fraction numerator 2 over denominator 3 left parenthesis straight x squared plus 2 right parenthesis end fraction dx
equals space 1 over 6 log left parenthesis straight x minus 1 right parenthesis minus 1 over 6 log left parenthesis straight x plus 1 right parenthesis plus 2 over 3 cross times fraction numerator 1 over denominator square root of 2 end fraction tan to the power of negative 1 end exponent open parentheses fraction numerator straight x over denominator square root of 2 end fraction close parentheses plus straight C
equals 1 over 6 open square brackets log left parenthesis straight x minus 1 right parenthesis minus log left parenthesis straight x plus 1 right parenthesis plus 2 square root of 2 tan to the power of negative 1 end exponent open parentheses fraction numerator straight x over denominator square root of 2 end fraction close parentheses close square brackets plus straight C

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