Integrate x^3/(x^5-1): Solutions

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SUMMARY

The integration of the function x^3/(x^5-1) yields a complex expression involving ArcTan and Log functions, as detailed in the provided solution. The answer includes terms such as 2 Sqrt[2 (5 + Sqrt[5])] and 4 Log[-1 + x], indicating the intricate nature of the integral. For practical purposes, utilizing Mathematica is recommended for solving such integrals, as manual computation is highly challenging and not straightforward.

PREREQUISITES
  • Understanding of integral calculus and techniques for integration.
  • Familiarity with functions such as ArcTan and Log.
  • Basic knowledge of algebraic manipulation involving square roots and polynomials.
  • Experience with computational tools like Mathematica for complex integrals.
NEXT STEPS
  • Explore advanced integration techniques in calculus, focusing on substitution and partial fractions.
  • Learn how to use Mathematica for symbolic computation and integration tasks.
  • Study the properties and applications of ArcTan and Log functions in calculus.
  • Investigate other complex integrals and their solutions to enhance problem-solving skills.
USEFUL FOR

Students and professionals in mathematics, particularly those studying calculus, as well as anyone interested in advanced integration techniques and computational tools like Mathematica.

Andy_ToK
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Hi, here is the question
integrate x^3/(x^5-1)
 
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Hi, here is the answer:

(2 Sqrt[2 (5 + Sqrt[5])]

1 - Sqrt[5] + 4 x
ArcTan[---------------------] +
Sqrt[2 (5 + Sqrt[5])]

2 Sqrt[10 - 2 Sqrt[5]]

1 + Sqrt[5] + 4 x
ArcTan[--------------------] +
Sqrt[10 - 2 Sqrt[5]]

4 Log[-1 + x] +

(-1 + Sqrt[5])

(-1 + Sqrt[5]) x 2
Log[1 - ---------------- + x ] -
2

(1 + Sqrt[5])

(1 + Sqrt[5]) x 2
Log[1 + --------------- + x ]) / 20
2Edit: The format didn't come out too well. Go to http://integrals.wolfram.com/index.jsp and do it yourself.
 
hi, thanks. I have the answer but wonder how to solve it without using calculator. Sorry if I wasn't clear.
 
Does it look like it would be easy?

My advice- leave it to Mathematica unless you have a pathological interest in doing near-impossible integrals (like GibZ).
 

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