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sauravshakya

  • 3 years ago

How to integrate this

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  1. sauravshakya
    • 3 years ago
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    |dw:1356849576104:dw|

  2. hartnn
    • 3 years ago
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    seems it doesn't have any closed form... http://www.wolframalpha.com/input/?i=integral+%28x%5E%281%2Fn%29-1%2Fx%5En%29%5En

  3. abb0t
    • 3 years ago
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    are we assuming n is a constant?

  4. malevolence19
    • 3 years ago
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    \[=\int\limits \left( \frac{x^{\frac{n^2+1}{n}}-1}{x^n} \right)^ndx\]

  5. malevolence19
    • 3 years ago
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    I see 2 approaches. Either do something like: u=x^(n^2+1)/n - 1 and solve for x in terms of u; then find dx, etc. Or try integration by parts by letting u=(...)^(n-1), dv=(...)^1dx; from there you could try to establish a pattern for n. But it is more than likely unable to be solved in closed form.

  6. wio
    • 3 years ago
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    @abb0t We know \(n\) is constant... but it would help to know if \(n\) is just a natural number or any real number.

  7. wio
    • 3 years ago
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    For some reason things like this want me to try binomial theorem, lol I'm just a bit crazy I guess.

  8. abb0t
    • 3 years ago
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    crazy

  9. wio
    • 3 years ago
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    If you assume \(n \in \mathbb{N}\) you can just use binomial theorem and get an answer in summation notation.

  10. wio
    • 3 years ago
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    Then if you're lucky, you might be able to simplify it from there, into something algebraic, but there is no guarantee it will happen.

  11. wio
    • 3 years ago
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    There is no reason to be afraid of summations when it comes to certain integrals... there is no guarantee the function isn't transcendental.

  12. malevolence19
    • 3 years ago
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    =\[\int\limits \sum_{k=0}^{\infty} \left(\begin{matrix}n \\ k\end{matrix}\right)(x^{\frac{1}{n}})^{n-k}x^{-nk}dx\] My only question with doing this is that in the binomial theorem the don't make mention of things of the form: \[(\alpha(n)+\beta(n))^n\] Where alpha and beta are numbers that depend on n.

  13. malevolence19
    • 3 years ago
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    I would assume it doesn't matter.

  14. malevolence19
    • 3 years ago
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    And that last x should have an (-1)^k also.

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