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How to integrate $$\int\frac{1}{x^{14}+1}dx$$ I've tried to use partial fraction decomposition but it doesn't work

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    $\begingroup$ Are you sure no bounds? $\endgroup$ – jamie Apr 23 '20 at 13:38
  • $\begingroup$ @jamie yep, I need to find indefinite integral of this $\endgroup$ – Limenal Apr 23 '20 at 13:40
  • $\begingroup$ the indefinite integral doesn't have a closed form, however if you are integrating in the whole $\mathbb{R}$ then you can use the residue theorem to find a closed form for it value $\endgroup$ – Masacroso Apr 23 '20 at 13:40
  • $\begingroup$ @Masacroso So, any "simple" methods (like from algebra) will not work? $\endgroup$ – Limenal Apr 23 '20 at 13:44
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The integrand is the sum of a geometric series with common ratio $-x^{14}$, so your integral equals

$$\int 1 - x^{14} + x^{28} - x^{42} + \cdots \; dx $$

$$= x - \frac{x^{15}}{15} + \frac{x^{29}}{29} - \cdots.$$

That's probably as good as you're going to do. (Plus constant.)

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