Integral of 1/(x^6+1)

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I+J = \int\frac{1}{1+x^6}dx##In summary, the individual has a homework problem that involves writing 1 + x^6 as a sum of two cubes and using partial fractions to solve it. They have attempted both methods but have gotten stuck on solving for the variables in the partial fraction decomposition. They are seeking guidance and have provided a Mathematica answer as a reference. They are also given a hint to break the integral into two parts and use a specific formula.
  • #1

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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  • #2
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.
 

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  • #3
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.
 
  • #4
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##
 

1. What is the formula for the integral of 1/(x^6+1)?

The formula for the integral of 1/(x^6+1) is ∫(1/(x^6+1)) dx = (1/6)tan^-1(x^2) + C, where C is the constant of integration.

2. What is the domain of the integral of 1/(x^6+1)?

The domain of the integral of 1/(x^6+1) is all real numbers except for x = ±i, where i is the imaginary unit.

3. How is the integral of 1/(x^6+1) calculated?

The integral of 1/(x^6+1) is calculated by using the substitution method or by using partial fraction decomposition.

4. What is the significance of the integral of 1/(x^6+1) in mathematics?

The integral of 1/(x^6+1) has significance in calculus and complex analysis. It is used to evaluate integrals involving trigonometric functions and can also be used in solving differential equations and in the study of complex functions.

5. Can the integral of 1/(x^6+1) be evaluated using numerical methods?

Yes, the integral of 1/(x^6+1) can be evaluated using numerical methods such as Simpson's rule or the trapezoidal rule. However, the accuracy of these methods may vary depending on the interval of integration and the number of subintervals used.

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