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$$\int_0^{\infty} \frac{x^8-4x^6+9x^4-5x^2+1}{x^{12}-10x^{10}+37x^8-42x^6+26x^4-8x^2+1}\,{dx} $$

The answer is $\pi/2$ (formula 34). How does one calculate this?

EDIT: I don't know complex analysis. I traced the problem back to a book but the book used complex analysis. I should have said that from the start, sorry.

Bmudtneduts
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    What is your background? Do you know complex analysis? – Ian Cavey Dec 24 '15 at 17:32
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    note that the integrand is equal to $$1/4,{\frac {-{x}^{5}-4,{x}^{4}-3,{x}^{3}+4,{x}^{2}+5,x+2}{{x}^{6} +4,{x}^{5}+3,{x}^{4}-4,{x}^{3}-2,{x}^{2}+2,x+1}}+1/4,{\frac {{x} ^{5}-4,{x}^{4}+3,{x}^{3}+4,{x}^{2}-5,x+2}{{x}^{6}-4,{x}^{5}+3,{x }^{4}+4,{x}^{3}-2,{x}^{2}-2,x+1}} $$ – Dr. Sonnhard Graubner Dec 24 '15 at 17:32
  • You asked a very similar question 4 hours ago and got a great answer. What's the point of this one? – Start wearing purple Dec 24 '15 at 17:39
  • @Startwearingpurple: That integral was also discussed some years back in this post. – Tito Piezas III Dec 24 '15 at 18:09
  • @I.Cavey, No, I don't know complex analysis. – Bmudtneduts Dec 24 '15 at 20:05
  • @Startwearingpurple I want to learn the techniques to solve these kind of problems. I tried the technique I was taught by that beautiful solution on this integral and many other things but none worked for me. – Bmudtneduts Dec 24 '15 at 20:11
  • The previous version of this Question also includes an Answer based on rational decomposition, although it is not perhaps as detailed as you would like. It might be best to keep everything in one place by adding a further Answer there rather than reopening this one. – hardmath Dec 24 '15 at 21:24

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