Can You Solve this Tough Indefinite Integral?

In summary, the conversation is about solving the integral \int\frac{1}{x^{2n} + 1}dx and the difficulties the speaker has encountered. They mention using de Moivre's theorem and factorising the denominator, but have not been successful in finding a solution. They also mention that the Mathematica online integrator gives a result in the form of a hypergeometric function, which they believe can be obtained through Taylor expansion. They also express a desire to find a simpler solution that does not involve Taylor or Maclauren methods.
  • #1
FedEx
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A long time since i posted at physics forums. Anyways, try helping me solve the following integral

[tex]\int\frac{1}{x^{2n} + 1}dx[/tex]

I tried many ways but all futile. The best way with which i could come up was factorising the denominator by de moivre's theorem. By finding the 2nth roots of unity. Hence i was able to express the denominator in a better way. But that's it. Dead end. I don't why but i get a feeling that we may able to do the sum by that way.

I am sorry that i am not able to presnt much work to you.

Hoping that you may be able to do the problem.
 
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  • #2
Well, according to the Mathematica online integrator, the result is a hypergeometric function. I believe you get this result by Taylor expanding in powers of x, doing the integral term by term, and then noticing that the result is a hypergeometric series.

http://integrals.wolfram.com
 
  • #3
Avodyne said:
Well, according to the Mathematica online integrator, the result is a hypergeometric function. I believe you get this result by Taylor expanding in powers of x, doing the integral term by term, and then noticing that the result is a hypergeometric series.

http://integrals.wolfram.com

Mathematica sometimes even gives answers in complex numbers to a simple problem. And the answer which i have in the book is not a hypertrigo function. There should be a way to bring the answer without Taylor or Maclauren cause we haven't been taught those yet.
 

1. What is a tough indefinite integral?

A tough indefinite integral is an integral that cannot be easily evaluated using basic integration techniques such as substitution or integration by parts. It often involves complex functions or multiple variables.

2. How do I solve a tough indefinite integral?

To solve a tough indefinite integral, you may need to use advanced integration techniques such as partial fractions, trigonometric substitutions, or specialized integration rules. In some cases, it may also be necessary to use numerical methods or computer software to approximate the solution.

3. Can a tough indefinite integral have multiple solutions?

Yes, a tough indefinite integral can have multiple solutions. This is because there may be different ways to manipulate the integrand or different values for the constants of integration. It is important to check the validity of each solution by differentiating and comparing with the original function.

4. How do I know if I have solved a tough indefinite integral correctly?

You can check if you have solved a tough indefinite integral correctly by differentiating your solution and comparing it with the original function. If they are equal, then your solution is correct. Additionally, you can also use computer software to check your answer or ask a math tutor for verification.

5. Can a tough indefinite integral be simplified?

Yes, a tough indefinite integral can be simplified in some cases. This could be achieved by using algebraic manipulations or applying integration rules such as the substitution or integration by parts. However, not all tough indefinite integrals can be simplified, and some may require more advanced techniques to evaluate.

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