How do I find the integral of 1/(x^6+1) using partial fractions?

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Homework Help Overview

The discussion revolves around finding the integral of the function 1/(x^6+1) using partial fractions. The problem falls under the subject area of calculus, specifically integration techniques.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to factor the expression by writing it as a difference of squares and expresses uncertainty about their approach to partial fractions. Some participants suggest alternative factoring methods, such as expressing 1+x^6 as a sum of cubes. Others provide hints involving breaking the integral into parts.

Discussion Status

The discussion is ongoing, with participants exploring different factoring techniques and approaches to integration. Some hints and suggestions have been provided, but there is no explicit consensus on a single method or solution yet.

Contextual Notes

Participants note varying levels of confidence in integration skills, and there is mention of using computational tools like Mathematica for verification, indicating a reliance on external resources for guidance.

freshman2013
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Homework Statement



basically the title

Homework Equations





The Attempt at a Solution


so I tried writing it as a difference of squares and got (x^3+1+sqrt(2)*x^1.5)(x^3+1-sqrt(2)x^1.5)
and I attempted partial fractions and I don't know if I did anything wrong, but then I got stuck when it came time to solve for the variables in the partial fraction decomposition. I'm not lost on this problem so If anyone has any clue, please guide me in the right direction. Thanks!
 
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I am not good at integrations... but here is a answer kind of thing done in mathematica...
you can check your results with it...
Sorry, could not really help you.
 

Attachments

Try writing [itex]1 + x^6[/itex] as a sum of two cubes:
[tex] 1+x^6 = 1 + \left(x^2\right)^3[/tex]

and factor, then apply partial fractions.
 
Hint :: ##\displaystyle \int\frac{1}{1+x^6}dx = \frac{1}{2}\int\frac{(1+x^4)+(1-x^4)}{1+x^6}dx##

and Break into two parts ##I## and ##J##
 

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